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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http : //books . google . com/ j ^. i -• i y _ I. i ^-_. ilL' i 1 ISdUi^Tiiq.oG.S'^O HARVARD UNIVERSITY LIBRARY OF THE Department of Education COLLECTION OF TEXT-BOOKS Contributed by the Publishers 3 2044 097 005 680 RATIONAL GRAMMAR SCHOOL ARITHMETIC BY GEORGE W. MYERS, Ph. D. PnOFESaOR OF THE TEACHING OF MATHEMATICS AND ASTRONOMY, SCHOOL OF EDUCATION, THE UNIVERSITY OF CHICAGO SARAH C. BROOKS CITY NORMAL SCHOOL, BALTIMORE, MD. CHICAGO SCOTT, FORESMAN AND COMPANY 1906 Harvard univerei^) DeptofEdueattonUbrary, (:iftcf^h'?P:;':li.sher: TR/^NSFFPPED TO HARVARD COLLEGE LIBRARY ^..u lb 1921 COPYRIGHT, 1903, BY BCOTT, FORESMAN AND COMPANV ROBERT O. LAW COMPANY. PRINTERS AND Bl N OERS, C H IC AGO PREFACE The present book is an outgrowth of the notion that arithmetic as a science of pure number, and arithmetic as a school science, must be treated from two essentially different stand- points. Viewed as a finished mental product, arithmetic is an abstract science, taking its bearings solely from the needs, of the subject; but viewed as a school subject, arithmetic should be an abstracted science, taking its bearings mainly from the needs of the learner. The former calls only for logical treatment, while the latter calls for psychological treatment as well. In other words, to be of high educational value the school science of arithmetic must take into full account the particular stage of the pupil's development. The abstract stage must be approached by steps which begin with the learner, rise with his tinfolding powers, and end leaving him in possession of the outlines of the science of arithmetic. To break vital contact with the learner at any stage of the unfolding process is fatal. A controlling principle in the development of the various topics of this book is that any phase of arithmetical work, to be of value, must make an appeal to the life of the pupil. But the social and industrial factors in American communities enter largely into the pupil's life. This renders material drawn from industrial sources and from everyday affairs of high pedagog- ical value for arithmetic. The recent infusion of new life into the curricula of elementary schools through the wide introduction into them of nature study, manual training, and geometrical drawing furnishes a basis for a closer unifying of the pupil's work in arith- metic with his work in the other school subjects. Wide use has been made of all these sources of arithmetical material. A rational presentation of the processes and principles of arith- metic can be secured as well through material representing real conditions as through material representing artificiail conditions. Not only have most of the problems been draven from real sources m IV . PBEFACE ' ■ but a very earnest effort has also been made to have all data of problems absolutely correct and consistent, to the end that infer- ences from them may be relied upon. With so rich a store as the book contains, however, it is perhaps too much to hope that no errors remain. The authors will deem it a favor to be notified of any errors that may be detected. It is well known that the majority of the pupils of the alementary school never reach the high school. Even these pupils, whose circumstances cut them off from advanced mathe- matical study, have a right to claim of the elementary school some useful knowledge of the more powerful instruments of algebra and geometry. For those who will continue their studies into the high school it is important that the roots of the later mathe- matical subjects be well covered in the soil of the earlier. The present book seeks to meet the needs of both classes of pupils through the organic correlation of the elements of geometry and algebra with the arithmetic proper. Treated thus, the geometry serves to illustrate the work of arithmetic and algebra, and the algebra emerges from the arithmetic as generalized number. In particular, this text aims to accomplish four main pur- poses, viz. : (1) To present a pedagogical development of elementary mathe- matics, both, as a tool for use and as an elementary science; (2) To base this development on subject-matter representing real conditions ; (3) To open to the pupil a wide range and variety of uses for elementary mathematics in common affairs — to aid him to get a working hold of his number sense; and (4) To give the pupil some training in ways of attacking com- mon problems arithmetically and some power to analyze quanti- tative problems. It is not intended that teachers should have their classes solve all the problems; but rather that each teacher should select such topics and problems as have a particular interest for his school. We trust that the suggestiveness of this book as to sources of problems and ways of handling them in the arithmetic class will be appreciated by many teachers. Such teachers will doubtless PBBFAOB V prefer to work out some topics more fully than is done in the text. It is thought that what is given in the text will make the fuller working out of special topics practicable and not difficult; The work makes continual call for estimating magnitudes and for actual measurement by the pupils. Let the children be sup- plied with measures, foot-rules, yardsticks, meter-sticks, etc., and encourage their constant use. Make so regular a feature of this work that pupils form the habit of estimating distances, areas, volumes, weights, etc., always correcting their estimates by actual measurement. All models, scales, or standards of measure made by pupils should be carefully kept and used in the later work. Pupils should also be supplied with instruments with which the exercises in constructive geometry may be actually done. A cheap pair of compasses is the only necessary purchase. It is better that the pupil make the rest of the apparatus needed. If for ally reason the pupil cannot actually do the constructive work it should be omitted altogether. The Introduction which precedes the work in the formal operations is a departure from current text-book procedure believed to be worthy of special remark. It consists of fifteen pages of simple, practical problems, relating to matters with which the pupil's experiences, in school and out of school, have familiarized him in an indefinite way, for the right understanding of which the use of numbers and of the arithmetical processes is necessary. This chapter, by furnishing to the pupil a gradual transition from his vacation experiences to the rather severe study of the formal processes- of arithmetic, by impressing the pupil with a sense of the real need and purpose of such study, thus preparing his inter- est for it, will be recognized as a deviation from common practice resting upon sound pedagogical grounds. Especial attention is invited to this introductory chapter; to the chapter on measurement, §§71-81, preceding and laying the foundation for common fractions; to the treatment of proportion, §§85-97, 115-118, 190, 191; to the lists of data for original prob- lems, §§29, 36, 44, 132; to the work in geography, §§23, 27, 54, 157; in farm account keeping, §35; on farm products, §62; in Vi PREFACE commerce, §28; in nature study, §§45, 122; 124, 155, 156, and elsewhere; in physical measurements, §§6-8, 125, 133; in paper- folding, §174; on the locomotive engine, §166; in constructive geometry, §§70, 168-180, 187; and in applied algebra, §206. Most teachers will approve dropping many of the time-honored but antiquated subjects and the curtailing of other topics of slight utility, and the introduction in their places of more timely and more real subjects. The many lists of data for original problems, §§29, 36, 44, 132; the treatment of longitude and time, §§181, 182, the descriptive work leading up to this topic; the extensive use of graphs to put meaning into arithmetical measures and to bring out the laws involved in numerical data, will commend themselves to teachers as useful and instructive means of keeping the number faculty employed on practical material. Another special feature is the numerous lists of problems that bear on the development of some important idea or law, having an interest on its own account. In these lists each problem is a step in a connected line of thought culminating in an important truth. This plan furnishes numerous problems, miscellaneous as to process, thereby requiring original mathematical thought, and still organically related to a central idea, thereby calling for the constant exercise of judgment. Examples of this may be seen in any section of the introduction; in the problems on geography, commerce, nature study, and elementary science, and also in the following sections: 35, 67, 68, 82, 83, 110, 115, 116, 118, 133, 134, 136, 155, 156, 157, 160, and in practically all the matter from p. 264 to the end of the book. For the maturity of pupils of the later grades this is believed to be an important feature. It avoids the danger, always present with lists of promiscuous problems when classified under the arithmetical processes to be exemplified, of reducing to the mechanical what should never be allowed to become mechanical, viz.: the analysis of relations. Throughout the book the instruction is addressed to the pupil's understanding rather than to his memory. But while accomplishing this, due regard has been had to the necessity of sufficient drill in pure number to enable the pupil to obtain both a conscious recognition of processes and considerable PBEPAOE YU facility in their automatic use. This is done out of the conviction that the fundamental arithmetical operations should be reduced to the automatic stage as early as possible, consistently with a clear understanding of them. The attention of teachers is called to the section on Short Methods and Checking at the close of the book. After pupils have clearly grasped the meaning of the arithmetical processes and have acquired some mastery of their uses, a relief from the tedium of long arithmetical calculations becomes a matter of great impor- tance. No one can become a rapid computer without short methods and, considering that the problems of daily life that call for arithmetical treatment do not have answers with them, no one can be certain of his results without means of checking calcula- tions. Every expert accountant uses them and the more expert he is the more does he use them. In fact, expertness consists very largely in the ability to shorten calculations and to apply rapid checks. Training in the use of short cuts and checks should con- stitute a much more important part of the pupil's work in arith- metic than is common. Constant use of these sections should be made through the seventh and eighth grades. Definitions, processes, -rules, and even special subjects are worked out under the guidance of the principle: "First its informal, though rational, use, and afterwards conscious recogni- tion of its formal use." Definitions of merely technical terms are given when called for by the development and where the need for them arises. Unusual attention has been paid to the numerous illustrations. The publishers have spared no pains to make them an important aid to the teacher in the development of the subject. None have been inserted for mere adornment and none are mere pictures of things or relations that are perfectly obvious to the pupil. Their aim throughout is to secure clearness, precision, and certainty of thought. It is thought the book may lay rightful claim to an innovation in this particular. For convenience in reviews and for reference, a synopsis of definitions and a full index are given. The authors' acknowledgments for suggestions and ideas are Vlll PREFACE due to numerous arithmetical writers and teachers. They desire to express their obligations in particular to Miss Katherine M. Stilwell of the School of Education, University of Chicago, for valuable help throughout the book ; to Mr. J. B. Eussell, Super- intendent of Schools, Wheaton, 111., and to Miss Ada Van Stone Harris, Supervisor of Primary Schools, Eochester, N. Y., both of whom read much of the proof. They desire most heartily to thank Mr. Stephen Emery of Lewis Institute, whose unceasing diligence and pains with the proofs have wrought distinct improve- ments on nearly every page of the book. If the book shall in some measure aid in putting the teaching of arithmetic on a more rational basis, thereby bringing about results more nearly commensurate with the time and energy put upon the subject in the elementary schools, the authors will deem their efforts repaid. The Authors. Chicago, August, 1905. TABLE OF CONTENTS PAGE Preface iii Introduction Scale Drawing 1 Reading Scale Drawing : 2 Cost of Living 3 Fencing a Farm 7 Areas of Fields ; . , g Physical Measurements (a) .... ^ 11 Physical Measurements (6) 12 Physical Measurements (c) 13 Wind Pressure 14 Dairying 16 Notation and Numeration 17 Digits 17 Place and Name Value of Digits 17 Periods 18 Decimal Notation 19 Reading Numbers 19 Writing Numbers 20 Roman Notation 20 Change of Notation 21 Addition — Definitions 21 Exercises 23 Problems 24 Distribution of Population in the United States 29 Measurements 31 Subtraction — Definitions 32 Exercises 34 Geography 37 Commerce 38 Numbers for Individual Work 4l Subtraction of Literal Numbers 42 Multiplication — Definitions 43 Tables 45 Current Prices 46 Farm Account Keeping 47 Problems 50 ix X TABLE OF CONTENTS PAGE! Multiplying by Factors. 51 Multiplying by 10, 100, 1,000, 10.000 .' 52 Multiplying by Numbers near 10, 100, 1,000, 10,000 52 Multiplying by 25, 50, 12i, 75, 500, 250 52 Multiplying when Some Digits of Multiplier are Factors of Others 53 Checking Multiplication 53 Multiplying by Fractional Numbers 54 Suggestions for Problems 54 Rainfall 56 Algebra 59 Division — Division and Subtraction Compared 60 Division and Multiplication Compared 61 Short Division 63 Applications 64 Long Division 65 Exercises 67 Larger Numbers 68 Geography 68 Division by Multiples of 10 70 Other Methods of Shortening Division 72 Tests of Divisibility 73 Checking Division 74 Town Block and Lots 76 House and Furnishings 78 House Plans 81 Applications of Cancellation 82 Farm Products of the United States in 1900 83 The Thermometer 85 Temperature Lines 86 Examples for Practice 87 Bills and Accounts — Exercises 88 Accounts 88 Problems of the Grocery Clerk 92 Family Expense Account 93 The Equation .*. 97 Constructive Geometry 98 Problems with Ruler and Compass 98 Adding and Multiplying Lines 100 Bisecting Lines 101 Drawing Angles 102 Exercises 103 Ornamental Figures 104 JIeasurement — Measuring Value 105 Measuring Length and Distance 106 TABLp OF C0NTEKT8 XI PAGE Measuring Surfaces 108 Development of a Room 109 The Parallelogram 110 The Triangle Ill Measuring Volume (Bulk) and Capacity 112 Measuring Weight 114 Measuring Temperature' 116 Measuring Time 118 Measuring Land 119 Plotting Observations and Measurements 120 ■ Heights and Weights of Boys and Girls 121 Measuring by Hundredths 123 Simple Interest 125 Common Uses op Numbers 127 Pressure of Air 127 Passenger and Freight Trains 128 Train Dispatcher's Report 130 Area of Common Forms 132 Problems 133 Introduction to Ratio and Proportion 135 Ratio ' 135 Proportion 137 Common Fractions 138 Fractions as Ratios and as Equal Parts 138 Reduction of Common Fractions 140 Factors, Prime and Composite 142 Greatest Common Divisor by Prime Factors 143 Problems 145 Fractions Having a Common Denominator 146 Fractions Easily Reduced to Common Denominators 146 Multiples 148 Finding the Least Common Multiple 149 Shorter Process for Three or More Numbers 150 Definitions and Principles 151 Addition of Fractions 153 Subtraction of Fractions 155 Multiplying a Fraction by a whole Number 157 Multiplying a Mixed Number by a Whole Number 159 Multiplying a Whole Number by a Fraction 160 Factors may be Interchanged 160 Multiplying a Fraction by a Fraction 162 Multiplying a Mixed Number by a Mixed Number 163 Dividing a Fraction by a Whole Number 165 Dividing a Mixed Number by a Whole Number 167 Xll TABLE OP CONTENTS PAOS Dividing any Number by a Fraction 168 Complex Fractions 172 Joint Effects of Forces 173 Exercises for Practice 175 Dividing Lines and Angles 176 Parallel Ruler 177 Exercises , 179 Uses of a 30° and a 45° Triangle 180 Drawing a Perpendicular 182 Scale Drawings of Familiar Objects 183 School House and Grounds 185 Proportion 186 Practical Applications 188 Decimal Fractions 190 Notation of Decimals 190 Numeration of Decimals 191 To Reduce a Decimal to a Common Fraction 192 Addition of Decimals (Rain and Snowfall) 193 Other Applications 194 Substitution of Decimals (Nature Study) 195 Stature and Weight of Persons 196 Pointing the Product of Decimals 198 Force Needed to Draw Loads on Road Wagon 199 Division by an Integer 200 Division by a Decimal 202 Problems 203 Ratio of Circumference of Circle to Diameter 204 Original Problems 205 Physical Measurements 206 Specific Gravity 207 To Reduce a Common Fraction to a Decimal 208 The Area of a Circle 210 Exercises for Practice 212 Composite Denominate Numbers — Defmi ions 213 Measures of Value 213 Measures of Weight, . 214 Measures of Length or Distance 214 Measures of Surface 215 Measures of Volume 215 Measures of Capacity 216 Measures of Time 217 Measurement by Counting 217 Exercises on Denominate Number Tables 219 Standards of Value 219 TABLE OF CONTENTS XIU PAGE Weight. , 4 220 Linear Measure 221 Surface Measure 221 Capacity 222 Time 223 Counting Things 224 General Exercises 224 The Metric System— Historical 229 Tables of Metric Measures 230 Metric and U. S. Equivalents 232 Percentage and Interest — Percentage , 235 Algebra 238 Gain and Loss 240 Meteorology 243 The Almanac 244 Geography 245 Commission 248 Trade Discount 249 Marking Goods 251 Interest 252 Algebra 257 Promissory Notes 259 Discounting Notes 260 Partial Payments 261 Applications to Transportation Problems — Locomotive Engine . . 264 Laws for the Drawing of Loads 269 Constructive Geometry — Problems 272 To Model a 3-inch Cube 274 To Model a Square Prism 275 To Model a Flat Prism 276 Comparison of Prisms \ 270 Volume of an Oblique Prism 277 Paper Folding 278 Perimeters 282 Quadrilaterals 283 Perimeters of Miscellaneous Figures 283 Measuring Angles and Arcs 285 The Sum and Difference of Angles 289 Products of Sums and Differences of Lines 293 Maps of the World 296 Locating and Describing Places on the Earth 297 Longitude and Time 300 Standard Time ; . . 303 Mensuration— Areas, Roofing, and Brick Work 306 XIV TABLE OF CONTENTS PAOB The Ti^apezoid 307 Development of Octagonal Tower 309 Land Measure 310 Volumes 312 Constructive Geometry 317 The Right Triangle 320 Squares and Square Roots 321 Cubes and Cube Roots 326 Similar Triangles 327 Uses of Similar Triangles 329 Applications of Percentage — Insurance 336 Taxes 338 Trade Discount 339 Stocks and Bonds 341 Compound Interest 343 Use of Letters to Represent Numbers —Problems 345 Uses of the Equation 348 Principles for Using the Equation 349 Problems 352 Statements in Words and in Symbols 354 Problems for Either Arithmetic or Algebra 356 Formal Work 359 Equations Containing Two Unknown Numl)era 360 Uses of the Equation — Thermometers 363 Applied Algebra 365 Equivalent Readings on tlie Three Thermometers 367 Problems 367 Laws of Thermometer Represented Graphically 368 Methods of Shortening and Checking Calculations Illustrations 369 Shortening and Checking Addition 370 Make-up Method of Subtraction 371 Shortened Multiplication 371 Shortened Division 373 Shortened Square Root; Square Root by Subtraction 374 Synopsis of Definitions 375 General Index 386 Answers • RATIOISTAL GRAMMAR SCHOOL ARITHMETIC DTTBODnCTION §1. Scale Drawing. oral work 1. The scale of the drawing of the picture frame is 1": 4", or i. How high is the frame? how long? 2. How long and how high is the place for the picture? Answer question 1 for a scale of j^f ; of t'j- t Bookoaae Q Mo|V|o UoUoUb 0o FltiUKB 1 3. What is the scale of a drawing in which a line 1 in. long represents 1 ft.? in which 1 in. represents 40 ft.? 100 ft.? 4. In Fig. 2, which is a scale drawing of a schoolroom, one- sixteenth of an inch represents i ft. How long is the room? the teacher's desk? How long and how wide are the pupils' desks? 5. In what direction do the pupils face, when seated at their desks? 6. Find by measurement how far it is from the northwest corner of desk 1 to the southwest corner of desk 4; from the west edge of desk 2 to the east wail of i;he room. 7. Make and answer other questions on this plan. 8. Make a similar (like) scale drawing from measurements of your schoolroom and the fixed objects within it. 9. Make a scale drawing from measurements of your school- house and grounds. Mo L b wo w p bo pU< p 7 [ [ I I I r-r-r -T tt i i i l 1 I I I I i I I T-m FIOITRK 2 RATIONAL GRAMMAR SCHOOL ARITHMETIC §2. Beading Scale Drawings. Using a foot rule marked to 8th8 and 16ths of an inch and noticing the scale, answer by measurement the problems on the drawing of figure 8. 1. How wide is the actual block that is shown by the drawing MNOPl How long is the block? M A B C 1 D s N K m "4zr Alley Vegetekble Garden. ^ m K W- -E S PleciLSure Gcxrdei\ Figure 3 SCALB F/ = 100'. 2. How wide is lot ^? B? A'? Z? 3. How long is the vegetable garden? How wide? 4. How wide is the pleasure garden? The southeast corner lot? The northeast corner lot? COST OF LIVING 3 5. The alley (25 ft. wide) is taken equally from the north and south halves of the block. How long is lot K^ The northeast lot? The southeast lot? 6. How long are the flower beds? How wide is the walk? 7. If a man cuta a strip along the west side of lot A with a lawn mower that cuts a swath 1 ft. wide, how many square feet of the lawn does he cut in passing along the side once? Twice? Ten times? How many strips must he cut to mow the entire lot? 8. How many square feet are there in lot A^ B? In the northeast corner lot? In the southeast comer lot? In lot K? 9. The baseball "diamond" on the northeast lot is a 90-ft. square. How many square feet are there within the lines con- necting the bases? 10. If a player strikes the ball and runs to third along the dotted lines, how far does he run? How far must a player run to make a "home run''? 11. The left fielder stands at L F^nd throws the ball to the pitcher at P. How far does he throw? 12. The center fielder at C F catches a ball which was struck at the home plate, throw:s it to third at T^ where it is caught and thrown to the home plate. How far has the ball gone? 13. How much will it cost to enclose the northeast quarter of the block with a fence costing 25 cents a foot? 14. Make and solve other problems on the drawing. 15. Notice the scale of any good maps in your geography and find by measurement about how far it is from Chicago to New York; to Washington, D. C. ; to New Orleans; from New York to St. Louis; to Pittsburg; to Boston; to San Francisco. §3. Cost of Livin;. oral wore 1. If the living expenses of a family are $70 per mo., what is the expense per year? 2. If house rent is $25 per mo. and other expenses are $50, what part of the total expense is house rent? 3. How many work days are there in a month? How many weeks in a year? EATIONAL GRAMMAR SCHOOL ARITHMETIC WRITTEN WORK 1. Find out at home or from your neighbor how much hard coal is required to supply some furnace or stove 1 mo. Supposing hard coal costs $7.50 per T., how much will the coal cost for the 6 winter months from November to May? 2. If soft coal costs $3.75 a T., and If T. goes about as far as 1 T. of hard coal, about how much would it cost to supply this fur- nace with soft coal for the 6 winter months? The prices given below are fair averages for a large city. They would be less in smaller places, and prices current in your com- munity are to be preferred to these. Grocer's Prices Apples pk. 1.35 Apricots can . 25 Baking powder. . . .J lb. can .20 Bread loaf .05 Butter lb. .28 Cabbage lb. 2c., head .10 Cauliflower head .15 Celery bunch . 10 Cheese lb. .15 Coffee lb. .30 Crackers lb. .10 Eggs doz. .25 Flour 25 lb. sack .50 Lemons doz. .25 Lettuce bunch .05 Lima beans can . 10 Milk qt. .08} Oatmeal 2 lb. pkg. .10 Olives pt. .30 Onions pk. .20 Oranges doz. .30 Peaches can .25 Pears can .25 Peas can . 121 Pickles doz. . 10 Potatoes pk. .25 Prunes lb. .10 Rice lb. .08i Note. — It will be assumed that small quantities. Grocer's Prices— Conhn«6d Rolls doz. 1.10 Salt lb. .05 Soap bar .05 Starch T lb .10 Sugar lb. .05 Tea lb. .60 Tomatoes can .15 Butcher's Bacon Chicken Fish, fresh Fish, salt Ham Lard Lobsters, shrimps, Mutton Oysters Pork Pork tenderloin . . . Porterhouse steak. Round steak Salmon Sausage meat Sirloin steak Spare ribs Turkey, duck, etc. Veal rates are the same for Prices lb. 1.20 lb. .14 lb. .121 lb. .10 lb. .18 lb. .13 etc... lb. .25 lb. .14 qt. .30 lb. .10 . .lb. .20 lb .18 lb. .10 lb. .20 lb. .10 lb. .16 lb. .08i lb. .121 lb. .14 either large or COST OF LIVING 5 Make such problems as : 3. If you purchase of a grocer 2 loaves of bread, 1 lb. of butter^ 1 lb. of coffee, and 1 pk. of potatoes, and give him $1, how much change should you receive, if prices are as quoted (named) in the table on the opposite page? 4. If you buy a 2-lb. sirloin steak and give the butcher 50^, what change should you receive, prices being as in the table? 5. Pupils may make out lists in the form of a statement and find the change due if the account is paid with $2, $5, or $10, thus : Bought • Paid Item Item 1. 2 doz. rolls @ 10c. $0.20 1. Cash $6.U0 2. 1 doz. eggs 25c. .25 3. 1 doz. oranges 30o. .30 4. 2 bunches celery 10c. .20 5. 1 lb. tea 60o. .60 6. 2 cans peas 12ic. .25 7. 1 lb. starch 10c. .10 8. 5 bars soap 5c. .25 9. 1 pkg. oatmeal 10c. .10 Total cost Balance due in change 6. Foot the statement above. 7. Make out a bill of supplies such as a hotel keeper might purchase at the grocer's or the butcher's. 8. Pupils may make and solve problems similar to the above, using local prices when convenient. 9. The actual living expense account of a family for January is as shown in the table. If this is a fair average monthly expense account for the fam- ily, what will be its living expense for one year? 10. The family pays $3000 for a home and is thereby saved $^6 jm Jan., 1902. Meat 118.50 Groceries . 27.75 Milk 1.88 Butter 2.60 Soft coal.. 9.65 Clothing . . 15.00 RATIONAL GRAMMAR SCHOOL ARITHMETIC mo. in rent. The yearly rent amounts to what part of the cost 6t the home? 11. If the home is in the suburbs of a city and the man must pay 10^ morning and evening 6 d. per wk. for car fare, how much does this add to his yearly expense account? The table given here shows the average yearly cost to one person for food, clothing, and household utensils from 1897 to 1901 inclusive: For Year Ending Bread- stuffs Meat Dairy & Garden Other Food Clothing Metals Miscel- laneous Totals Dec.31, 1897 Dec.31, 1898 Dec.31, 1899 Dec.31, 1900 Dec.31, 1901 $13.51 13.82 13.25 14.49 20.00 $7.34 7.52 7.25 8.41 9.67 $12.37 11.46 13.70 15.56 15.25 $8.31 9.07 9.20 9.50 8.95 $14.65 14.15 17.48 16.02 15.95 $11.57 11.84 18.09 15.81 15.38 $12.11 12.54 16.31 15.88 16.79 Sums Averages . . . Note. — In the above table breadstuff s include wheat, corn, oats, rye, barley, beans, and peas ; meat includes lard and tallow ; dairy and garden Eroducts include vegetables, milk, butter, eggs, and fruit; the miscel- meous articles include a variety of things which make a part of the cost of living for the average family. 12. Fill out a column like the *Hotals" column, thus finding the total cost of living for 1 person for each of the 5 years. 13. What was the increase in cost of living from 1897 to 1901? 14. Has the'cost of any of the items decreased during this period? How much has been the change in cost in each case? 15. What fractional part (what part expressed as a fraction) of the whole cost of living for the year is the cost of breadstuflfs in 1897? in 1899? in 1901? 1st Arts. 4i|i = mi. Note. — The average of any five numbers is J of their sum. 16. Find the average yearly cost of living for the 5 years. 17. Find the average cost of each item (breadstuffs, meat, etc.). 18. Fill out a column like the "totals" column, with the sums of the numbers in the second, third, fourth and^fth lines. What does each of the totals mean? 19. Give the change, from year to year, in the total cost of foods for one person by finding the difference between each total and the one next after it. FENCING A FARM §4. Fencing a Farm. — A farm was divided into fields and seeded as shown in the drawing (Fig. 4). The scale of the drawing is 1 in. to 80 rd. This means that 1 in. in the drawing represents 80 rd. in the farm, and that all other lines longer or shorter than 1 in. represent distances in • the farm proportionately longer or shorter than 80 rd. ORAL WORK 1. How long is the corn- field north and south? how wide? 2. How long and how wide is the wheatfield? the north oatfield? 3. How long and how wide is the meadow? the south oatfield? 4. The drawing of the — i p road is ^V i'^- wide; how wide is the road? 5. How long and how wide is the farm? 6. How long must a wire be to reach entirely round the farm? 7. How many rods of barbed wire will be needed to inclose the farm with a 3- wire fence? how many miles (320 rd. = 1 mile)? WRITTEN WORK 1. How many rods long is the partition (division) fence across the farm from east to west? what part of a mile is its length? How many rods of wire will make it a 4-wire fence? 1st Ans. IGO. 2. How many rods of wire will run a 4-wire fence across the farm from the middle of the north side to the middle of the south side? Ans. 640. 3. How many rods of wire will run a 4-wire fence between the wheatfield and the north oatfield? between the meadow and the south oatfield? 4. How many bales of wire of 100 rd. each will be needed for these cross fences and division fences? AnSo 19.2. .Figure 4 8 RATIONAL GKAMMAK SCHOOL ARltflMETlO 5. How much will the wire cost for the 3-wire inclosing fence @ $3.50 a bale of 100 rods? Ans. $67.20. 6. The posts for all the barbed wire fences are set 1 rd. apart. Hoff many posts will be needed to fence around the farm? 7. How much will these posts cost @ 25^ apiece? 8. How much will the wire cost @ $2.60 a bale for the two 4-wire cross fences, and the two division fences between the wheat and the north oatfield and between the meadow and south oat- field? 9. How much will the posts for these fences cost @ 15^ apiece? 10. Find the total cost of wire and posts for these fences. 11. The house and barn lot is separated from the pasture by a 5-board fence. The boards used are -^^...^;' ■" '" ""■^;:::i=^;:;;;^ ^ 1 in. thick, 6 in. wide, and 12 ft. 7' ^^E^'^^Z^^:^3!F ll! long. How many boards will reach L^S^^^^^^^^^^^^E ' ^"^® along the north side of the " ' "''^^ ~^^J^ house and barn lot? Ans. 55. once along the east side of the lot? 13. How many boards will be needed for the 5-board fence along both sides? 14. The posts are set 6 ft. between centers, that is, a post is set every 6 ft. (Fig. 5). How many posts will be needed for the north side? for the east side? for both? Ut Ans. 110; 2d Ans. 109. 15. How much will these posts cost @ 23^ apiece? Lumber is sold by the hoard foot. A board foot is a board 1 ft. long, 1 ft. wide, and 1 in., or less, thick. IG. How many board feet of lumber in a board 1 in. thick, 1 ft. wide, and 5 ft. long? the same thickness and width and 10 ft. long? 12 ft. long? 17. How many board feet in a board 1 in. thick, 10 ft. long, and 2 ft. wide? the same thickness and length and G in. wide? 9 in. wide? 18 in. wide? 1 in. wide? 8 in. wide? 18. How many board feet in a board 12 ft. long, 4 in. wide, and 1 in. thick? 2 in. thick? AREAS OF FIELDS 9 19. How many board feet in a *'two by four" scantling 10 ft. long (2" X 4" X 10')* ? 16 feet long? 20. How many board feet in a "four by four" scantling 10 ft. long? 12 ft. long? 18 ft. long? 21. How many board feet in a fencing plank 1" x 6" x 12'? in 5 such planks? 22. How much will the fencing lumber of problem 12 cost @ $18 per M board feet? 23. What will be the total cost of the lumber and posts for the fence on the north and east sides of the lot? 24. It cost 8^ apiece to have the post holes bored, and it took 4 days' work by 3 men @ 11.50 per day to build the fence. Three patent gates costing $12 apiece were put in the fence. Find the total cost of the lot fence, including gates. 25. What was the total cost of all the fencing done on the farm? §5. A^reas of Fields. oral wore 1. The whole farm contains ICO acres; how many acres are there in the cornfield? in the wheatfield? in the meadow? in the south oatfield? in the pasture? in the lot around the house and bam? 2. What is the width of each of these fields in rods? 3. If the meadow were divided by a north and south central line, how many rods wide would each half be? How many acres would there be in each half? 4. How many square rods in a strip 1 rd. wide and 80 rd. long? 5. How many square rods in an acre? 6. How wide must a strip of land be to contain an acre if it is 80 rd. long? 40 rd. long? i mile long? 7. Compare the sizes of the wheatfield and the north oatfield ; of the cornfield and the wheatfield; of the meadow and the wheat- field; of the meadow and the south oatfield; of the meadow and the house and bam lots; of the lots and the pasture; of the pasture and the south oatfield. ♦Tbe ms^Tk (') means foot or feet; (") means inch or inches. 10 RATldXAL GRAMMAR SCHOOL ARITHMETIC WRITTEN WORK 1. What would be the income per acre from the farm planted in corn, if the average yield were 47 bu. per acre and were sold at 28^ per bu ? What would be the gross (total) income from the whole farm for this year? 2. If the farm were rented @ $5 per acre, how much would remain for the tenant * from each acre? from the entire farm? 3. What would be the owner's income from the whole farm, not allowing for expenses? 4. Which is the more profitable way for the owner to rent his farm, @ $6 per acre cash, or for i of all the crop delivered to mar- ket, supposing that the whole farm is planted to corn and that a yield of 48 bu. per acre and a price of 30^ per bushel can be obtained every year? Note. — ^Rent of the first sort is called "cash rent," of the second sort, "grain rent." 5. If this same farm will produce 20 bu. of wheat per acre and a price of 60^ per bushel can be obtained for it, which is the more profitable crop to the owner, wheat or com? how much more profitable? 6. Compare the profits to the owner of a grain-rented farm from an oats crop of 40 bu. per acre and a price of 22^ per bushel with the profits of the wheat crop of problem 5 ; also with corn crop of problem 4. 7. In problem 5 which would bring the larger income to the owner, and how much, cash rent @ $5 per A., or grain rent @ i delivered? 8. Answer the same question for the oats crop mentioned in problem 6. 9. A cornfield of 68 acres produced an average yield of 48 bu. per acre. How many bushels did the farm yield? 10. It cost $5.85 per acre to raise the crop. At 46^ per bu. what was the net value of the crop? 11. If the tenant paid cash rent @ 15.50 per acre, what was his net profit from the crop? How much did the owner receive? «The tenant is the farmer who raises the crop on another man*s farm. PHYSICAL MEASUREMENTS 11 Figure 6 §6. Physical Measurements (a)* — A tin can, or pail, 18" high and 10" in diameter is filled nearly full of water. A second can, or pail, 9" to 12" high and 9" in diameter is inserted, mouth downward, within the water. «^ A smooth hole should be punched in the p bottom of the smaller can. A rubber tube passed under the edge of B the inverted can and stretched over the end ^ of a piece of glass tubing, which is held in an upright position inside the inverted can by light cross-braces, may be used to get air. into the can. If a cork, which fits into the hole at A, is drawn, the inner can sinks readily in the water; after which the cork is inserted air- tight. A foot rule, marked with eighths or sixteenths of an inch, stuck with putty to the side of the inside can, permits the read- ings to be taken. Such an apparatus, when used as in the problems below, is called a spirometer. A more elaborate instrument is shown in Fig. 6. Common spools may be used for the pulleys that the cords pass over, and sand may be pressed into the cans w, w, until they just balance the weight of the inverted can or pail. 1. If the inside diameter of the inverted can is 8.8 in., for each inch the inner can sinks into the water, there will be 60.85 cu. in. of water inside the can, the stopper at A being removed. How many cubic inches (capacity) of water will there be in the can when it has sunk 2 inches? 5 inches? 8 inches? Note. — If a pail is used, the corresponding number of cubic inches is found by measuring the distance in inches across the mouth of the pail, multiplying one-half this distance by itself, and the product by 3|. 2. How many cubic inches of water will there be in the can when it is sunk in the water only 1^ inches? If inches? 2^ inches? 2^^^ inches? 3. Let the can now be pushed down as far as it will go and the stopper at A be pushed in tightly. Read the scale. When a pupil blows through the tube until the inside can rises 1 inch, how 12 RATIONAL GRAMMAR SCHOOL ARITHMETIC many cubic inches of air has he expelled from his lungs into the can? IIow many when the scale indicates that the can has riseD I inch? lyV inches? 1} inches? ly^ inches? If inches? 4. Fill your lungs full and blow into the tube as long as you can without danger. Suppose you raise the can 2", what is the capacity of your lungs in cubic inches? 5. The average lung capacity for a man 5 ft. 8 in. tall is 204 cu. in. This average capacity may be called the normal lung capacity. How much does your lung capacity fall short of the normal value for a man? 6. The normal lung capacity in cubic inches for a man is 3 times his height in inches. For a woman, it is 2.6 times the height. Divide your lung capacity by your height in inches, and compare your quotient with these numbers. 7. The chest measure of a man should be not less than half hie height. How much greater or less is your chest measure than it should be according to this law? §7. Physical Measurements (b). 1. Fill out in your notebook a record like the one below, which is a copy from a student's notebook : Name, John Morrow; age, 10 yr.; height, 52 in.; weight, 61 lb. r on Inspiration (breathing in) S8 in. Lung capacity, Chest measures, ] ^^ Expiration (breathing out) ^4i in. 104 cu. in. * (^Mean,* 26\ in. Height in inches by 3 (if a boy). Result 53 X S = 156 cu. in. Height in inches by 2.6 (if a girl). Average to each inch of height, 2 cu. in., for 104. -*- 5f = ;?. Chest average = 26\ in. Half height = 26 in. 2. How much do your measures exceed or fall short of the normal values? 3. Do your measures show that you need chest exercise? 4. How many in the class or room are above or below the normal? * The mean of 2 numbers is half their sum. PHYSICAL MEASCREMEXTS 13 5. Compare class or room averages* to see whether the average for class or room is about normal. 6. Make the comparison of problem 5 after a course of exercises in physical training, and record the changes due to the exercises. 7. Every month or two during the year find results like these called for in problems 2 and eo; keep your results, and show what changes are taking place in your measures. g8. Physical Measurements (c). — The following table gives the average height in inches and the average weight in pounds for boys and for girls year by year from 4 to 15 years of age. The num- bers are the averages of careful measures of heights and weights of hunflreds of boys and girls in the schools of Chicago, Boston, Cincinnati, and St. Louis. These averages may be called normal values for boys and girls of school age : Height Yearly Growth Weight Yearly Growth AGS Boys Girls Boys Girls Boys Girls Boys Girls 4vr. 39.2 in. 39.0 in. 87.31b. 35.31b. 5- 41.6 '• 41.4 " 40.6" 39.7 " 6 " 43.7 •' 43.3" 44.7" 43.0 " 7 " 45.8 " 45.5 " 48.7 " 47.0 " 8 •* 47.9 ** 47.6 " 58.8 " 52.0 " 9 ** 49.7 " 49.5 " 58.7 " 57.1 " 10 " 51.6 •• 51.3 " 64.8 " 62.2 " 11 " 53.5 ** 53.4 ** 70.1 " 68.1 " 12 - 55.2 '* 55.8 " 76.7 " 77.4 " 13 " 57.3 " 58.5 " 84.9 " 88.4 " 14 " 60.0" 60.2" 95.0 " 98.3 " 16 •' 62.4 - 61.3 " 106.5 " 105.0 " 1. Subtract each number in the column of heights from the number next beloio it and find the growth in height for boys and for girls from year to year. Do the same for the weights. 2. When is the yearly growth in height greatest for boys? for girls? What is the yearly growth in each case? 3. Make and solve similar problems for weights. 4. How much do your height and weight exceed, or fall short of, the normal values for your ageV *Tbe claims average is the sum of the measures for all pupils iu the class divided by tbe number of pupils. 14 RATIONAL GRAMMAR SCHOOL ARITHMETIC 5. How do the averages for your room compare with the values in the table for the same years? 6. How much do your height and weight exceed or fall short of the averages for pupils of your own age in your room? Wind Pressure. ORAL WORK 1. Which is easier, to walk against or with the wind? 2. If you are walking against the wind with a raised umbrella, why do you have to push forward harder than if you are walking with the wind? 3. If you hold a board 1' wide and 10' long flatwise against the wind, how hard does the board push against you if the wind pushes against each square foot of the board with a force of 2 pounds? 3 pounds? 5 pounds? 2^ pounds? 20 ounces? 32 ounces? WRITTEN WORK The wind is a stream of air moving along with a certain speed, or velocity. The greater the speed, or velocity^ the stronger does it push against objects it strikes The velocity of wind in miles per hour and the pressure per square foot are as given here: Light breeze 3| mi. .75 oz. Moderate breeze .. ^ ** 3.33 '* Fresh breeze IGJ '* 1 lb. 5 ** Stiff breeze 32J ** 5 *' 3 '^ Strong gale 56^ ml. 15 lb. 9 oz. Hurricane 79J " 31*' 4 ** Violent hurricane 97 J " 46 " 12 ** Note. — Use measures from your own schoolhouse or from other build- ings in your neighborhood in preference to the numbers in the problems. 1. What is the total pressure in pounds on the west side of a house 30 ft. by 25 ft., due to a light breeze from the west? to a stiflf breeze? to a strong gale? 2. What is the pressure on the side of a tall building 265 ft. by 80 ft., due to a hurricane blowing squarely against it? 3. Find the wind pressure against the side of a load of hay 25 ft. long and 12 ft. high, due to a strong gale blowing squarely against it. WIND PRESSURE 15 4. Find the number of pounds pressure against a signboard fence 100 ft. by 14 ft. , due to a strong gale blowing squarely against it. 5. What is the wind pressure tending to over- turn a square chimney 30 ft. wide at the base, 15 ft. wide at the top, and 175 ft. high, due to a .violent hurricane blowing squarely against one of the flat faces? Note. — To obtain the area of the effective surface against which the wind is blowing, multiply the height of the chimney by its breadth halfway up. This breadth is the half sum of the widths at the top and bottom (Fig. 7). 6. Find the wind pressure on the side of a passenger coach 70 ft. long and 15 ft. high, due to a stiff breeze blowing squarely against it : (a) If the wind blows against the whole rec- tangle 15' x 70' ; (b) If the wind is not obstructed by the strip 4^^' X 70', from the track to the bottom of the coach box and running the length of the coach. 7. Find the wind pressure on the side of a train of 10 such cars as that of problem 6 (^), due to a strong gale, blowing squarely against it. 8. Find the wind pressure on a diamond- FlGUBE 7 FiGUBE 8 shaped signboard, due to a violent hurricane blowing squarely against it, if the diamond is 12' long (AB = 12') and 6' broad at the middle (CD = 6'). (See Fig. 8.) 10 RATIOXAL GRAMMAR SCHOOL ARITHMETIC §10. Bairying. — The following table is an extract from the milk record of the University of Illinois herd of milk-cows : WEIGHT OF MORNING (A.M.) AND EVENING (P.M.) MILKINGS IN POUNDS 1901 Queen Beauty Beech- wood Mtbtlk Spot ROSE Not. 1 TOTAI. a.m. 6.6 p.m. 6.5 a.m. p.IXL a.ixL p.m. a.m. p.m. a.m. 12.0 P.J11. 11.5 a.in. 11.0 p.m. 9.6 11.0 10.4 12.5 10.2 7.3 5.7 2 5.6 6.0 11.0 8.4 11.0 11.0 7.2 6.6 11.5 10.5 11.8 8.8 3 6.4 6.0 10.8 9.5 11.7 11.5 7.3 5.7 12.0 10.4 11.2 9.8 4 6.0 6.5 12.5 10.2 12.6 13.0 7.2 6.0 11.5 11.5 11.0 9.8 5 6.0 7.5 11.4 10.2 12.0 11.4 6.7 5.8 11.6 11.3 11.0 10.2 6 6.6 6.8 12.0 10.0 8.0 11.6 8.0 5.0 12.5 11.8 11.7 10.0 7 7.4 8.5 10.8 12.4 12.0 11.7 7.0 7.2 12.0 12.5 11.6 12.0 8 7.8 7.3 11.2 11.6 13.2 12.2 7.2 5.7 11.5 11.3 11.5 10.1 9 7.5 6.4 11.6 10.4 12.0 11.6 6.6 5.4 12.0 12.2 10.7 10.0 10 7.8 7.2 11.0 10.4 12.0 12.0 6.7 4.7 11.5 8.5 11.8 10.6 11 6.8 7.3 12.2 11.0 13.6 11.5 7.7 5.3 9.0 10.0 12.3 10.0 12 7.8 7.7 12.5 12.0 13.0 11.6 8.0 5.5 12.0 12.6 12.2 10.6 13 7.6 8.0 13.4 11.5 13.5 13.0 7.2 5.8 11.1 13.0 11.8 112 14 7.0 8.0 12.0 11.8 14.0 12.7 7.7 5.7 12.6 12.7 11.5 10.3 15 8.8 7.5 13.0 10.7 15.0 12.0 7.6 5.0 14.0 12.4 12.3 9.0 16 9.0 7.0 12.5 11.0 15.2 13.7 7.0 5.5 12.7 12.3 14.1 8.5 17 8.0 7.0 12.5 10.5 14.0 13.0 6.1 6.4 10.0 9.0 12.0 10.5 18 8.1 7.7 13.4 10.9 14.0 11.5 6.6 5.0 12.5 11.5 11.0 9.1 19 7.0 7.0 11.5 11.4 13.6 11.0 6.2 5.3 12.2 11.7 11.1 10.0 20 7.5 6.5 12.6 10.5 12.7 12.0 6.5 4.9 11.8 11.4 11.5 9.5 21 7.0 7.6 12.0 10.2 13.7 11.3 6.1 5.2 13.0 11.0 10.6 9.6 22 8.0 7.5 12.2 11.9 14.7 11.2 6.7 6.0 12.4 10.7 11.3 9.5 23 7.5 9.0 12.2 11.0 14.4 12.0 7.0 5.5 12.8 10.8 11.3 10.4 24 7.0 7.1 13.5 10.6 13.8 11.5 7.2 5.1 13.0 10.7 12.0 10.4 25 7.4 8.0 12.5 11.2 14.0 10.5 7.4 5.7 12.7 11.7 12.0 11.0 26 6.5 8.4 12.0 11.8 12.0 12.0 6.1 5.8 13.2 11.5 11.5 10.5 27 7.5 8.5 13.7 11.8 14.0 13.0 6.1 5.8 13.0 12.7 11.4 10.5 28 7.2 9.2 12.0 12.0 13.6 18.8 6.6 49 13.1 12.5 11.7 ro.5 29 6.5 9.0 13.5 12 6 15.0 13.5 6.7 5.0 10.2 12.6 12.5 12.6 30 7.2 8.0 11.'; 12.7 14.2 12.2 6.0 4.3 12.4 11.5 12.7 10.3 1. How many pounds of milk did Queen give the first week of Nov., 1901, at the morning milkings? at the evening milkings? at both milkings? 2. At which milking did Queen give most milk and how much more than at the smallest milking? 3. Make and answer similar questions for the remaining weeks and for any or all of the 6 cows. NOTATION AKB ITOMEEATION §11. Digits.— 0, 1, 2, 3, 4, 5, G, 7, 8, 9. 1. How many characters are given above? By what names do you know the first character? It is called zero^ or naught. These ten characters are called digits or figures; and by means of them all numbers, are expressed. §12. Place and Name Values of Digits. 1. Read the following numbers: $30 3 30 30,000 300 mi. .3 300 300,000 3 yd. .03 3000 ^^ What is the same in all these numbers? What is different in the numbers? 2. Read: 7 feet 7 days 7 hundreds 7 millions 7 tenths How many feet? 7 what? How many hundreds? 7 what? How many millions? 7 what? In all these cases, what is the same? What is different? 3. In the number CGG, what does the first G on the right mean? the second G? the third? 1st Ans, G ones or G units. 4. If the value which depends on the name of a digit is called its name value^ what shall we call the value which depends on its placef 5. How many values has a digit? What gives the first value? the second? Which is the permanent (fixed) value? Which is the changeable value? §13. ORAL WORK 1. Give the place value of 2 in each of the following numbers : 20 2 200 .2 20,000 2000 2222 2. In 286, for what number does the 2 stand? the 8? the 6? 17 18 RATIONAL GRAMMAR SCHOOL ARITHMETIC We may read the number 2 hundreds, 8 tens, 6 units, or two hundred eighty-six, meaning two hundred eighty-six units. Writing the names of the three places, they appear: I •a fl S3 » 2 8 6 § g fl » S P meaning 200 + 80 + 0. 3. Read the following numbers, first as hundreds, tens, and units, and then as units: 729 91G 335 494 845 642 538 739 . §14, Periods. — For convenience in reading, the digits of large numbers are grouped into periods of three digits each. These periods have names depending upon their position; but the three places in each period are always hundreds, tens, and units.* The first seven periods are units, thousands, millions, billions, trillions, quadrillions, and quintillions. The first three periods are sufficient for common purposes. w w <n (D w 00 go t II I I I I §g? §g^ §gi Is? §g? li^ §g? wSp whp wSp wSp ffiSp wSp w§p 253 462 571 324 918 645 284 WRITTEN- WORK Write the following in figures : 1. Forty-five thousand, two hundred eighty -four ; six thousand, nine hundred thirty-four; three hundred twenty -one thousand, one hundred thirteen ; five thousand, five. Note. — Fill all vacant places with zeros. *In Great Britain a notation different from that explained in $14 is used. The method of §14 is called the French notation and is used altogether in the United States, France, and Germany. The English notation groups the digits into periods of six figures each : Billions Millions Units 865324,786593,279368 In the English notation a mlUion millions make a billion, a million billions a trillion, etc. In the French notation a thousand millions make a billion, a thousand billions a trillion, etc. NOTATION AKD KtJMERATlOK 19 2. Nine thousand, ten; ten thousand, nine; six hundred thou- sand, six hundred; five million, fifty thousand, three ; eighty thou- sand, eighty. §15. Decimal Notation. oral work 1. 10 equals how many units? 1 equals what part of 10? 2. 1 hundred equals how many tens? how many units? 1 ten equals what part of 1 hundred? 1 unit equals what part of 1 hun- dred? 3. 1 thousand equals how many hundreds? 1 hundred equals what part of 1 thousand? 4. The number denoted by each digit in 1111 is how many times as great as the number denoted by the digit to its right? The number denoted by each digit equals what part of the num- ber denoted by the digit to its left? 5. How many tenths in 1 unit? 1 unit equals how many times 1 tenth (.1)? 6. 1 hundredth (.01) equals what part of 1 tenth? 1 tenth equals how many times 1 hundredth? 7. How does the place value of the unit change from right to left? from left to right? §16. Beading Numbers. 3,007 .6 365,834 .27 4,004,240 1.07 2. Read and tell the meaning of numbers of the tables on pp. 30, 39, or 40. 3. Read aloud the following: Mean radius of the earth = 636,739,510 centimeters Mean distance to moon = 76,429,120 rods Mean radius of sun = 138,560,000 rods Mean distance to sun = 92,897,500 miles Mean distance to sun = 149,500,000,000 meters Light travels in one minute = 11,199,000 miles 20 RATIONAL GRAMMAR SCHOOL ARITHMETIC 4. The table gives the number of vessels engaged in the trade of the chief countries of the world for 1901, and the number of tons they could carry. Eead some of the numbers: Country Greece Austria-Hungary Denmark The Netherlands Japan Spain Sweden Russia Italy France Norway Germany The United States... The United Kingdom NUMBEB Steamers Sailing Vessels 150 243 377 318 511 489 672 659 381 850 866 1,350 794 7,906 1,078 366 1,200 1,004 2.035 1,112 2,328 3,372 1,931 2.595 2,913 2,560 4,614 15,219 Tonnage Steamers Sailing Vessels 139,187 294,960 256,631 330,132 338,354 460,193 314,572 347,875 435,426 547,895 507,158 1,561.078 916,153 7,944.095 320,797 322,897 387,727 451,949 510,175 562.068 607,868 850,695 947,079 961,259 1,893.096 2,106,895 2,318,276 10,304,338 §17. Writing Numbers, written work Write in numerals the following : 1. Four billion, two hundred thirteen million, one hundred twenty-two thousand, six hundred seven; forty-five billion, four hundred fifty million, three hundred twenty-seven thousand, one. 2. Seven hundred eighteen billion, two hundred million ; one hundred ninety thousand, four hundred two; three million, one hundred twenty-four; fifteen thousand, nine; ninety; one hun- dred eighty-six billion; one hundred forty-seven million, one hun- dred thousand, one hundred. The digits, excepting zero, were first used by the Hindus, but they were introduced into Europe by the Arabs, and we call the numbers written with these digits Arabic numerals, §18. Roman Notation. — The Romans used letters instead of digits to represent numbers. This notation is still used upon monu- ments, and in numbering chapters and volumes of books. In the Roman notation, 1= 1 L= 50 D= 500 V= 5 C = 100 M = 1000 X = 10 ADDITION 21 When a letter of less value, as I, is placed before a letter of greater value, as'V, the value of the less is to be taken from that of the greater; thus, IV means 4 and XC means 90. When a letter of less value follows one of greater value, its value is added to that of the greater; as, VI for 6, XV for 15, and CX for 110. Repeating a letter repeats its value ; as, XX for 20, CC for 200. A horizontal line over a letter increases its value a thousand- fold; as, C meaning 100,000, D meaning 500,000. To interpret (change to Arabic notation) numbers written in the Eoman notation, always begin on the right. §19. Change of Notation, writtek work 1. Change from Roman to Arabic : XCV CMXIX CDLXXXIV DXXV MLXXX MDCCLXXVI DCIV MCCLXX LXIX XOIX DLVI OCXI 2. Change from Arabic to Eoman : 10 65 100 572 619 1902 30 77 500 78 584 1774 60 80 140 50 400 1565 49 95 46 246 2000 2590 ADDITION §20. Definitions. oral work 1. A bicyclist rides 19 mi. Monday and 10 mi. Tuesday; how far does he ride both days? 2. I paid $28 for a suit of clothes and $20 for a bicycle; how much did I pay for both? 3. A four-sided lot has sides of the lengths: 25 yd., 4 yd., 11 yd., and 28 yd. ; how far is it around the lot? 4. A man works for me 7 hr. one day, 6 hr. the next day, 7 hr. the next, and 8 hr. the next; how many hours does he work for me in all? 22 RATIONAL GBAMMAK SCHOOL ABITHXETIO 5. A newsboy sold 18 papers Wednesday, 12 papers Thursday, and 16 papers Friday; how many papers did he sell in all? 6. The number of hours the sun shone on each of the days of a certain week was: 8 hr., 6 hr., 3 hr., 7 hr., 5 hr., 9 hr., 2 hr. How many hours of sunshine were there during the week? 7. A farmer's fields are of the following sizes : 40 acres, 80 acres, 16 acres, and 34 acres. How many acres in all? 8. How many dollars are $25 and $35 and $20? 9. How many feet are 12 ft., 8 ft., 6 ft., 12 ft., and 4 feet? The answer to problem 1 is the same as the answer to the question: "What single distance is just as long as the distances 19 mi. and 10 mi. combined?" Answering problem 2 answers the question: "What siugle amount equals $28 and $20 combined?" Combining numbers into a single number is addition. The result of addition is called the sum or amount. The numbers to be combined or added are the addends. Thus in the problem 29 mi. Sum 19 mi. and 10 mi. are the addends and 29 mi. is the sum. The sign of addition is +. It is read "plus." In some cases it may be read "and," though it is better to use the correct read- ing, "plus," at all times. The sign = when written between two numbers means that they are equal. It should not be read "is" or "are," but always "equal" or "equals." 10. Eead and answer these questions : (1) 3 + 8 + 4 = ? (4) 12+ 7 + 8 = ? (7) 3 + 15 + 4 + 9 = ? (2)7 + 6 + 8 = ? (5) 6 + 13 + 7 = ? (8)6 + 16 + 8 + 9 = ? (3)9 + 8 + 5 = ? (6) 5 + 18 + 9 = ? (9)5 + 14 + 9 + 7 = ? 11. In these questions, what numbers should stand in place of the question mark? (1) 2 8's + 5 8's = ? 8's. (5) 9 y's + 8 y\ = ? y's. (2) 6 9's + 7 9's = ? 9's. (6) 15 z'% + 10 z'^ = ? z'^. (3) 12 13's + 7 13's = ? 13's. (7) 8 rt's + 17 a's = ? a's. (4)* 7 a;'s + 5 a;'8 = ? re's. (8) 16 J's + 20 ^'s = ? ^'s. * Read 4 thus, "7 x's plus 5 aj's equal how many a's?" ADDITION 23 Exercises. written work 1. In five trips a street-car carried the following number of passengers: 30, 42, 45, 60, 65. Find the whole number of pas- sengers. When the sums can not be held in memory, it is convenient to arrange the addends in this form : Convenient FOBM Solution. — Writing the addends so that units, tens, . hundreds, etc., shall stand in the same vertical column, 30 we add the units first; thus, 5, 10, 12 units = 1 ten and 42 2 units. Write the 2 units in units column under the line 45 and the 1 ten in tens column. Then, 6, 12, 16, 20, 23 tens 60 = S hundreds and ^ tens. Write the 3 in tens column 65^ and the 2 in hundreds column. Then add the partial 12 sums as shown. 23_ 242 2. On these trips the conductor collected $1.50, $2.10, $2.25^ $3.00, $3.25. The fares for the five trips amounted to what? 3. In one week my street-car fare was 35^, 25^, 40^, 25^, 30^, 30^. Find the whole amount. 4. In one evening a telegraph operator sent messages contain- ing the following numbers of words: 2, 6, 3, 12, 6, 3, 3. What was the total number of words dispatched? 5. Fred bought a note book for 10^, a lead pencil for 5^, a foot rule for 10^, a compass for 25^, and a bottle of ink for 10^. What was the amount of his bill? 6. A train moves the following distances in five successive hours: 40 mi., 45 mi., 50 mi., 50 mi., 48 mi. What distance is traveled in the given time? Find the total sales in each of the following problems: 7. Silk sales: ^ yd., 3^ yd., J yd., %\ yd., 3 yards. 8. Linen sales: 2^ yd., 5 yd., 6 yd., 2| yd., 5 yards. 9. Ribbon sales: 1^ yd., i yd., 3 yd., 2 yd., 12 yards. 10. Thread sales : 25 spools white cotton, 6 spools black cot- ton, 3 spools black linen, 4 spools A silk, G spools twist. 11. Handkerchief sales: 6, 12, 3, 6, 12, 6, 3, 2. How many dozen were sold? 24 EATIONAL GRAMMAB SCHOOL ARITHMETIC 12. Kotion sales : 6 papers pins, 2 papers safety pins, 6 papers needles, 2 cards darning wool, 2 combs, 12 bunches tape, 4 bunches braid. How many articles were sold? 13. Coffee sales: i lb., 2 lb., 2^ lb., 3 lb., J lb., 1^ lb., 5 lb., 4 pounds. 14. Sugar sales: 18 lb., 12 lb., 12 lb., 20 lb., 5 lb., 6 pounds. 15. Flour sales: 50 lb., 150 lb., 26 lb., 15 lb., 25 pounds. 16. Potato sales: 2 bu., 1^ bu., 4 bu., 6 bu., 2 bu., 1^ bu., 3 bushels. 17. Apple sales: 2 bu., 1 bu., 3 pk., 2 pk., 1 pk., 2 bu., 1^ bu., 4 bushels. (Answer in bushels.) 18. Egg sales: 7 doz., 3 doz., G doz., 4 doz., 8 doz., 5 doz., 2 dozen. 19. Butter sales: 6 lb., 3 lb., 3 1b., 2i lb., H lb., 3 1b., 5 pounds. §22. Problems. , gfS^ ► t 1 . 1 1 i i Playground « § Garden i \ FIGUBS ! 1. A fifth grade staked out a rectangular plot of ground to be used for a playground and school garden. The eighth grade pupils fenced the plot in and ran a cross fence separating the parts. The entire plot was 126 ft. wide by 275 ft. long (126' X 275'). How many feet of fence were needed? Solution. — Arrange the addends in convenient form. Beginning at the bottom of units column, think the sums thus, 12, 17, 23, ^8 equals 2 tens and 8 units. Write the 8 in units column. Add the 2 tens to the tens digits, thinking (not speaking) 4, 6, 18, 15, £^ tens, equals ^ hundreds and £ tens. Write the 2 tens in tens place and add the hundreds to the hundreds in the third column, thus 8, 4, 6, 7, 9 hundreds. Write the 9 in hundreds place, and since we are adding feet the sum is 928 ft. Test the correctness of the sum by beginning at the top of each column and adding downward. If the sum is the same the work is probably correct. This is called check- ing the work. 2. Add 187, 892, and 478. Convenient Fobm 275 ft. 126 ft. 275 ft. 126 ft. 126 ft., 928 ft. Addends Sum ADDITION" 25 3. In a certain school there were enrolled one month : 96 children in the kindergarten 100 in the fifth grade 182 in the first grade 95 in the sixth grade 143 in the second grade 83 in the seventh grade 133 in the third grade 76 in the eighth grade 123 in the fourth' grade How many pupils were in the whole school? 4. During the same month, the numbers of cases of tardiness were as follows : Kindergarten... 10 Third grade 8 Sixth grade 12 Eirst grade 12 . Fourth grade 5 Seventh grade.. 4 Second grade... 4 Fifth grade 9 Eighth grade... 2 What was the total number of cases of tardiness? 5. The numbers of cases of absence for the same month were as follows, beginning with the kindergarten: 87, 60, 46, 21, 15, 20, 15, 14, 9. Find the total number of cases. Teachers may take reports of their own or of other schools and make similar problems. 6. Fill out for each of these cities the total number of rains during May and June, 1902, and the total rainfall in inches. Do not rewrite the numbers. Crrr Number or Rains Rainfall in Inches MAT June Total MAY June Total Chicacro 11 13 14 13 13 10 15 16 18 15 16 13 4.46 4.57 3.46 5.03 2.83 2.21 6.00 6.76 6.16 4.19 7.29 9.08 Des Moines Detroit Kansas City Omaha St. Louis. The length in feet, the number of officers and men, the dis- placement in tons, and the indicated horse power of eight of the first-class battleships of the United States navy are as given in the table on the following page. 26 RATIONAL GRAMMAR SCHOOL ARITHMETIC Name Length Men Displace- ment HORSB POWJBR Alabama Wisconsin . . . . Kearsarge^ . . . Kentucky Iowa Indiana Massachusetts Oregon Total . . . 368 368 368 368 360 348 348 348 585 585 520 520 444 465 424 424 11.525 11,525 11,525 11,525 11,340 10,288 10,288 10,288 11,866 10,000 11,954 12.318 12.105 9,738 10,403 11,111 7. If these 8 vessels stood in a straight line with their ends touching, how far would they reach? 8. How many officers and men are needed to man the 8 ships? 9. The displacement is the number of tons of water the vessel pushes aside as it floats. What is the combined displace- ment? 10. What is the combined horse power of the engines of the 8 ships? 11. At the close of 1900 the numbers of teachers, schools, and pupils in the 6 provinces of Cuba were as follows : Province Teachers Schools Pupils Havana Puerta Principe. Santa Clara Santiago Pinar del Rio. . . Matanzas Total 904 246 879 645 274 619 904 246 879 645 274 619 41.383 9,355 44,872 33,983 13,282 29,398 Find the totals and tell what they mean. 12. The 6 European countries from which most immigrants came to the United States in 1901 were as follows: ADDITION 27 COUNTHY Immigrants in 1901 Immigrants in 1900 MALE Female Total MALK Female 1 Total Italy Austria Hungary Russia 106,306 78.725 54,070 12,894 12,875 29,690 34,665 31,187 17,667 10,456 76,088 79,978 60,091 16.672 10,262 24,047 34,499 31,066 19,058 8,388 Ireland Sweden Total Fill out all the totals and tell what they mean. 13. The published daily circulation of a city newspaper from week to week is here given : First Week Second Week Third Week Fourth Week Fifth Week DAY Copies "1... 255, 572 2... 303,062 3... 323, 513 4... 312, 724 5... 306,009 6... 297,918 Day Copies 8... 305, 725 9... 303,489 10... 304,991 11... 304. 746 12... 305, 515 13... 299,095 Day Copies 15... 304.636 16... 302, 173 17... 302, 650 18... 304, 255 19... 302.942 20... 297, 684 Day Copies 22... 804. 836 23... 304,698 24... 310.870 25... 299, 780 26... 801,455 27... 298,446 Day Copies 29... 303,383 30... 302.005 Total Total Total Total Total Total for mo. Find the weekly totals and the total for the whole month. 14. In the year 1891 the United States imported 94,628,119 lb. of coffee, which was 54,2*62,757 lb. less than was imported during the two previous years. How many pounds were imported during the two previous years? 15. The United States imported 79,192,253 lb. of tea in 1889; 83,494,956 lb. in 1890; and 82,395,924 lb. in 1891. How many pounds of tea were imported in the three years? 16. In 1889 $27,024,551 worth of molasses was imported into the United States; in 1890 $31,497,243 worth; and in 1891 $2,659,172 worth. How many dollars' worth was imported during the three years? 28 RATIONAL GRAMMAR SCHOOL ARITHMETIC 17. Of the yearly trade within New York state, $1,650,000,000 worth of the freight passes over the railroads; $150,000,000 over the canals; and $250,000,000 over Long Island Sound and the lakes. What is the total value of this trade? The Ten Largest Cities of the United States Population New York 3,487.202 Chicago 1,698,575 Philadelphia 1.293,697 St Louis 575.236 Boston 560,892 Baltimore 508,957 Cleveland 381,768 Buffalo 352,387 San Francisco 842,782 Cincinnati 325,902 The Ten Largest Foreign Cities Population London 4,433,018 Paris 2,536,834 Canton 2,000,000 Berlin 1,677,304 Vienna 1,364,548 Tokyo 1,268,930 St. Petersburg 1,267,023 Pekin 1,000,000 Moscow 988,610 Constantinople 900,000 18. Find the total population of the ton largest cities of the United States. Test the correctness of your addition hy adding the columns from top downward. Teachers may omit the following method of checking if thought too difficult. A very useful method of checking long problems in addition is known as casting out the nines. To cast out the nines from a number add its digits and whenever the sum equals or exceeds 9, drop 9 and continue adding the next digits to what remains, dropping 9 whenever the sum equals or exceeds 9. The last remainder is called the excess. Illustration.— Cast the 9's out of 647,255. Beginning on the left, 6 + 4=10; drop 9, giving the remainder 1. 1 + 7 + 2 = 10, drop 9. 1 + 5 + 5 = 11, drop 9. The last remainder is 2 and this is the excess of 647,255. To check addition by casting out nines, cast the nines out of the addends and the sum. Then cast out the nines from the excesses of the addends. If this last excess of the Excesses equals the excess of the sum, the work is probably correct. To obtain the excess of the excesses of the addends: 5 + 6=11, drop 9, giving 2. 2 + 6+5 = 13, drop 9, giving 4. 4 + 1 = 5, the excess of the ex- cesses of the addends. Since this equals the excess of the sum, the work is checked. Illustration EXCESS 8,465 3,282 4,497 2.957 7,642 5 6 6 5 1 26,843 5 = 6 5 = excess of the sum. 5 = excess of excesses of addends. ADDITION 29 19. Find the population of the ten largest cities of Europe and Asia. (See table, p, 28.) 20. Find the total population of these twenty cities. 21. Find the population of those cities in both lists having more than 900,000 inhabitants. 22. Make and solve similar problems based on the table. Distribntion of Population in the United States in 1900. FIGUBB 10 On the map the main geographical divisions are surrounded by heavy full lines:—. The divisions are separated into sections by heavy dotted lines: . Thus the New England states are the eastern section of the North Atlantic division ; and New York, Pennsylvania, and New Jersey are the western. Teachers may select such problems from this list as have a special geographical interest to their school. 1. From the table of numbers given on the following page, find the total population of the New England states for 1900. 2. Of the North Central division. 3. Of the Western division. 4. Of the United States with outlying territory. 30 Rational orammar dcHOOL arithhetio 5. Find the total area in square miles of the United States with outlying territory. Area and Population op States and Tbrb ITORIBS State or Tbbhitobt AREA IN SQ. MILBS Popula- tion 1900 Pupils IN Elem. & SBa Schools State or Tebbitort area IN SQ. Miles Popula- tion 1900 Pupils in £lbii. & Sec. Schools Me N. H Vt Mass. .... R. I Conn N.Y N.J Penn 33,040 9,305 9.565 8,315 1.260 4,990 49,170 7,815 45.215 694,666 411,588 843.641 2,805.346 428.666 908.420 7,268,894 1.883.669 6.802.116 130,918 65.193 66.964 474.891 64,637 155.228 1,209.574 316.055 1,151.880 Ky Tenn, ... Ala Miss La Tex. Okl Ark Ind.T.... 40,400 42,050 62,250 46.810 48,720 265,780 39,030 53,850 31.400 2.147.174 2,020.616 1,828,607 1,661,270 1,381,625 3.048,710 398.331 1,811.564 892,060 501.893 485.354 376.423 360.177 196.169 678,418 314.662 99,602 Nortb At- lantic divi- sion South Cen- tral divi- sion Del Md D. C Va W.Va.... N. C S.C Ga Fla 2,050 12.210 70 42.460 24.780 52,250 30.570 59,475 68.680 184.736 1.188.044 278,718 1,854,184 958,800 1,893,810 1,340,316 2,216,331 628,542 33.174 229,882 46,619 858,825 232,343 400.452 281,891 482,678 108,874 Mont. . .. Wy Col N. M Ariz Utah Nev Idaho Wash. ... Ore Cal 146,080 97.890 103.926 122.680 113.020 84,970 110,700 84.800 69,180 96.030 158,360 243.329 92.631 639,700 196,310 122,931 276,749 42,835 161,772 518.103 418.536 1,485,058 39,430 14,512 117,555 36.736 16,504 73,043 6,676 36,669 97,916 89,406 269,786 SoTitb At- lantic divi- sion Western division .... Ohio Ind Ill Mich Wis Minn. ... Iowa Mo N.D S. D Neb Kan 41,060 36.360 56.650 58,915 56.040 83,365 56,025 69,415 70,795 77,650 77.510 82,080 4,157,645 2.516,462 4,821.650 2.420,982 2,069,042 1,751,394 2.231,853 3,106.665 319,146 401,570 1,066,800 1,470,495 829. ICO 564.807 958.911 498,665 445.142 899,207 554,992 719,817 77 086 96.822 288,237 389,588 U. S. with- out outlying territory... Alaska ... Hawaii... Phil. Is.. Tutuila . . Guam . . . Porto \ Ricof 590.884 6,449 114.410 77 150 3,531 (i8.592 164,001 6.961,339 6,100 9,000 963,243 North Cen- tral divi- sion Total CT. S. with outly- ing territory i 1 .1 ADDITIOK 31 6. Find the area in square miles of the North Central division. 7. Of the South Central division. Which has the greater territory? 8. What state in the Union contains the smallest number of square miles? the greatest? What state supports (contains) the greatest population? the smallest? How do the areas of these two states compare? 9. Find the number of pupils attending school in the New England states ; in the North Central division. 10. Find the total number of children attending school in the United States. 11. Make such original problems as these: Find the total population in 1900 of the 13 original states; the total area of the 13 original states; of the states east of the Mississippi river, etc. 12. Check the correctness of your work in problems 2 and 7 by adding the columns first as a whole and then adding the foot- ings of the separate sections and comparing the two sums. 13. Find the population and the area of the eastern North Atlantic states ; of the western. 14. Find the number of pupils attending the elementary and secondary schools in both the sections mentioned in problem 13. 15. Check the work of problems 1 and 4 by casting out the nines. §24. Measurements. A square unit, or a unlt^^ square, is a square each of whose sides is 1 unit long. The area of a surface is the number of square units in it. ' 1 1. What is a square inch? a square foot? a figure u square yard? a square mile? a square rod? 2. If a rectangle is made up of 5 rows of 10 square units each, what is its area? How many tens of square FiGUM 19 iinits are in the rectangle? 32 RATIONAL GRAMMAR SCHOOL ARITHMETIC 12 3. If a rectangle is 12 units long and 6 units wide, how many twelves of unit squares are there in the area? 4. If a rectangle __ is a; units long and Fegubb 13 . . I , 9 units wide, how many a;'s of unit squares are in its area? Note. — Seven x's is written 7a? and is read Figubb 14 **8«ven a;." 5. How wide must a rectangle z inches long be to contain 6« sq. in.? 152; square inches? 0. How long must a rectangle 15 in. wide be to contain 15a square inches? 7. The area of one rectangle is Sz sq. in. and of another 7z sq. in. ; how many square inches in their sum? 8. How many 9's are 8 9's and 7 9's? How many JI2's are 8 I2's and 7 12's? How many 2;'s are Sx and 7x? Sz + 7z = ? 9. Add these numbers : (1) 9a (2) 8a (3) 2G:c (^) 48z (5) 76y (6) 695 8a 6a Vtx 98« 49^ 79* (7) 6wi (8) (ySn (9) 84c (10) TM (11) imx (12) 960a; 8w Ibn 76c 39fZ 319a; 869a; 9m 98m 37c 79rf 876a: 48a; SUBTRACTION . Definitions. ORAL WORK §26. 1. A man earns $180 a month and spends $140. What is the difference between his earnings and expenditures? 2. One farmer owns 320 A. of land, another 80 A. Find the difference in size of the two farms? 3. From a cheese weighing 43 lb., 23 lb. were sold; how many pounds remained? 4. From a bin containing 72 bu. of oats, I use 32 bu. How •jiany bushels remain? SUBTRACTION 33 5. The sum of two numbers is 48, and one of the numbers is 28 ; what is the other? 6. Wliat is required in the first two problems of this section? in the second two? in the fifth problem? 7. What is subtraction? What is the result in subtraction called? Subtraction is the way of finding: (1) The difference of two numbers expressed in the same unit, (2) The remainder after taking a part of a number away from the whole number. The number from which we subtract is the minuend. The number to be subtracted is the subtrahend. The result is the dif- ference^ or remainder. Subtraction is also the way of finding either one of two addends when their sum and the other addend are known. The known sum is the minuend^ the known addend is the subtrahe7idy and the unknown addend is the difference^ or remainder. The sign of subtraction — is read "minus," and when placed between two numbers means that their difference is to be found. The minuend is always written before the sign. 8. Tell what number the letter stands for in these problems: (1) 8 -5 = a; (3) l(S-l = y (5) 15-8 = ;? (2)17-9 = rc (4)25-y = 8 ((;)lG-;2; = 8 9. What digit should take the place of the question mark iii these problems? (1) 7 8'8-5 8's = ?8's (4) 9:c-(;.T = ?a: (2) 8 12's - 3 12's = ? 12's (5) 15 x - 1 x^'i x (3) 4 a; - 2 a; = ? :c (G) 15 a; - ? a; =» 8 a: WRITTEN^ WORK 1. A traveler has a journey of 437 mi. to make and he has already traveled 199 mi. of it; how much farther must he travel? Solution.— 437 means 400 + 30 + 7, and 199 means 100 + 90 + 9. We arrange the numbers conveniently, thus: Minuend 437 mi. = 300 mi. + 120 mi. + 17 mi. Subtrahend 199 mi. = 100 mi. + 90 mi. + 9 mi. Remainder 238 mi. = 200 mi. + 30 mi. + 8 mi. Beginning on the right, 9 units can not be taken from 7 units. We take one of the 3 tens and add it to the 7 units, giving 17 units. Then 17 units — 9 units = 8 units. Write a' in units column. Passing to tens 34 RATIONAL OIlAMMAR SCHOOL AElTHMMlO column, we can not take 9 tens from the 2 tens remaining, so we take one of the 4 hundreds and add it to 2 tens, making 12 tens. Then 12 tens — 9 tens = 3 tens. Write 3 in tens column. Finally, 3 hundreds — 1 hundred = ^ hundreds. Write the fJ in the hundreds column. The Vemainder is 238 miles. 2. How can yoa prove 238 to be the correct remainder or difference in problem 1? 3. The minuend is 1284, and the subtrahend is 876. What is the difference, or remainder? Prove. 4. When the water was dried out of 38 oz. of natural soil, the dry soil weighed 29 oz. Make a problem from these facts, solve it, and prove your solution correct. 0. A subtrahend is 99, and the difference is 338. What is the minuend? IIow is it found? 6. From 72 hr. subtract 1 da. and 4 hours. Note. — A correct answer to this problem would be 72 hr. — 1 da. and 4 hr. But the answer is to be expressed in a single unit, or name. To do this, first change 1 da. and 4 hr. to hours. 1 da. = 24 hr. ; 24 hr. + 4 hr. = 28 hr. 72 hr. — 28 hr. = 44 hours. 7. From 30 ft. take 5 yd. What must first be done to obtain the result in a single unit? Make complete statements. 8. 20 qt. - 2 pk. = x qt. What is the first thing to be done? Find the value of x. 9. From J yd. take | yd. What is the unit of J yd.? of f yd.? What then must be done before subtracting? Why? Note. — Only numbers expressed in the same unit can be subtracted if the result is to be expressed in a single unit. 10. Write the last problem, using the sign of subtraction. §26. Exercises. 1. From a barrel of vinegar containing 31| gal., suppose the following quantities to be removed, and give successive remainders: G gal. ; 2i gal. ; 5 gal. ; 3 gal. ; 2^^ gal. ; 2^ gallons. 2. From a wagon box containing 42 bu. of potatoes, a farmer sold the following quantities: 2 bu,, 4 bu., 3 bu., 2 bu., 5 bu., 6 bu., 9 bu., 4 bu. How many bushels remained? 3. On March 31 the gas meter read 30,000 cu. ft., and on April 30, 45,000 cu. ft. How many cubic feet of gas had been used durii^g the month? StTBTRACTIOK 35 4. A horse and his rider weigh 1375 lb. The man weighs 160 lb. What is the weight of the horse? 5. Find and read the differences in these exercises: (1) (2) (3) $9274.75 $4246.38 1,024,637 yd. 1846.19 ' 1571.16 725,448 " (2) $4246.28 1571.16 (2) $3472.06 2189.24 6. Find the differences : (1) (2) (3) 6,420,014 rd: $3472.06 1228f 1,382,741 '' 2189.24 936^ 7. During one week a merchant's transactions (or business dealings) at his bank were as follows: deposits (moneys put in), $217.40, $343.27, $290.00, $365.18, $380.24, $415.17; with- drawals (moneys taken out), $200, $320.25, $195.75, $286, $240.16, $395.25. Which item was the greater at the close of the week, and how much? 8. During one week of Xovember, 1901, 63,188 cattle were received at the Chicago stock yards; during the corresponding week of 1899, 32,722 were received. What was the increase? 9. Great Salt Lake is 4200 ft. above sea level, and Lake Su- perior 602 ft. X is the difference in elevation. Find value of x. 10. Cuba contains 45,884 sq. mi., and the Philippines, 114,410 sq. mi. Find the difference in areas. For the numbers for problems 11-17, see the tables^ p. 36. 11. Find, the total receipts at Chicago for the week Xov. 25-30, 1901, of cattle; of calves; of hogs; of sheep. 12. Find the total shipments. 13. Find the difference between receipts and shipments. 14. Compare the receipts with those of the previous week; with those of the corresponding week of 1900; of 1890. 15. Compare shipments in the same manner. 10. Find total receipts for one week in the four markets, Chi- cago, Kansas City, Omaha, and St. Louis, of cattle; of hogs; of sheep. 17. Compare Chicago receipts with those of tjie other cities. 36 EATIONAL GRAMMAR SCHOOL ARITHMETIC A Week's Receipts and Shipments op Live Stock. Chicago, Nov. 30, 1901 RECEIPTS Cattle Calves Hogs Sheep Monday, Nov. 25 16.078' 6.941 17,583 5.857 800 537 735 481 162 40 39,630 41,654 51,275 44.090 25,000 26,170 i 18,297 12,393 12.462 2,000 Tuesday, " 26 Wednesdav. ** 27 Thursday. ** 28 (Holiday) . . Friday, '* 29 Saturday, " 30 Total 63.188 53.965 32,722 3,814 1.822 1,335 273.426 191,104 159,724 104.528 55,046 61,718 Previous week Corresponding week 1900 Corresponding week 1899 shipments Cattlb Calves Hogs Sheep Mondav Nov 25 2.578 1,904 5,647 1,875 400 2 1 16 205 30 7,543 1.495 5,712 4,308 5,000 2,a52 4,518 4,501 2,127 1,000 Tuesdav. ** 26 Wednesday^ ** 27 Thursday, ** 28 (Holiday) . . Friday, " 29 Saturdav. *' 30 Total 18,699 18.438 9,556 324 352 210 29,486 23.675 16,637 24,429 14,832 3,182 Previous week Corresponding week 1900 Corresponding week 1899 Live Stock Receipts for One Week at Four Markets Cattle Hdbs Sheep Chicago Kansas City Omaha St. Louis Total Previous week Corresponding week 1900 Corresponding week 1899 Corresponding week 1898 Corresponding week 1897 47.300 29,300 16,800 10,700 201,600 89,000 65,200 38,100 71,300 14,900 12,800 5,800 144.600 111.500 91.700 128.400 143,000 481,700 339,800 206.100 365.500 374,100 173,600 85,000 81,300 95.600 105,600 SUBTRACTION" . . ^'^ 18. Compare the total receipts of the week with those of the previous week; of the corresponding week of 1900; 1899; .1898; 1897. §27v Geography. 1. Lake Titicaca is 12,645 ft. above the level of the sea, and Lake Superior 602 ft. Find the difference in elevation. 2. Mt. Everest is 29,002 ft. above sea level, and Mt. Blanc 15,744 ft. What is the difference in altitude? 3. The Mississippi basin has an area of 1,250,000 sq. mi. ; the Amazon, of 2,500,000 sq. mi. How much greater is the basin of the Amazon than that of the Mississippi? 4. The area of the Philippines is 114,410 sq. mi. ; that of California, 158,360 sq. mi. The state is how much larger than the islands? 5. Compare the same islands with Texas; with Illinois; with Rhode Island. (See table, p. 30.) 6. Of a population of 6,302,115 in Pennsylvania, 1,151,880 are pupils in elementary and secondary schools. How ma-ny per- sons are not attending those schools? 7. In a certain year the Hawaiian Islands produced 221,694 T. of sugar, while Cuba produced 880,372 T. How much more did Cuba produce than the Hawaiian Islands? 8. In 1890, 25,403 manufacturing establishments in Xew York City paid for wages $230,102,167; for materials, $366,422,- 722; for miscellaneous expenses, $59,991,710. The value of the products was $777,222,721. AVhat was the total profit for all the establishments? ' 9. The battle of Lexington was fought in 1775, and the battle of Santiago in 1898. How many years elapsed between the two battles? For the numbers for problems 10-19, see table, p. 30. 10. What is the difference of the areas of the largest and the smallest states? 11. How many more persons are in the state with the largest population than in the one with the smallest? 38 RATIONAL ORAMMAB SCHOOL ABITHKETIG 12. How many more children attend school in one of these two states than in the other? 13. Make and solve other problems, comparing the areas, total populations, and school populations of the different states. 14. Which is the larger, and by how much, the North Atlantic division or the South Atlantic division? the North Central or the South Central? the South Central or the Western? 15. Compare the total populations of these groups. 16. How many {arsons are not attending elementary or second- ary schools in Maine? in Mississippi? How do these two results compare? 17. Find how many persons are not attending these schools in your own state ; in bordering states. 18. How many more persons are not attending these schools in New York than in Illinois? How does this result compare with the difference in the total populations of these states? 19. Make and solve other problems of interest to your school. 20. Find, by comparing the populations for 1890 (p. 39) with those for 1900 (p. 30), the increase of population for the ten years in each state of the North Atlantic division ; in the whole division. 21. Find the increase of population in your own state from 1890 to 1900. What is the average yearly increase? 22. How many years since Illinois was admitted as a state? since your native state was admitted into the Union? 23. Make similar problems from the table opposite and the table on p. 30, solve your problems, and check your work. 24. Each principal division of states is separated, into two or more sections by heavy dotted lines on the map, p. 29. Find how much greater, or less, the growth of population since 1890 has been in your section than in the other section, or sections of the same division. §28. Commerce. 1, Find the total export trade of the United States with the twelve countries given in the table on p. 40, in 1891 and in 1901. 2. Find the increase, or decrease, for each country, and make a list of these results. Whenever the difference is an increase write + before it. When it is a decrease write - before it. Make the subtractions without rewriting the numbers, recording results. fitJBTRACTIOK 3d Dates of Admission and Populations for 1890 of States and Territories Geographically Grouped Date op POPULA- Date of Popula- Admission TION 1890 Admission tion lim Me Mar. 15, 1820 661.086 Ky June 1, 1792 1,848,635 N. H June 21, 1788 376.530 Tenn June 1, 1796 1.767.518 Vt Mar. 4, 1791 332.422 Ala Dec. 14, 1819 1,513,017 Mass Feb. 7, 1788 2,238.043 Miss Dec. 10, 1817 1.289,600 R. I May 20, 1790 345.506 La April 30, 1812 1,118,587 Ck)nn. . . . Jan. 9, 1788 746,258 Tex Dec. 29. 1845 2,235,528 N. Y July 26. 1788 5,997,853 Okl May 2. 1890 61,834 N.J Dec. 18. 1787 1,444,933 Ark June 15, 1836 1,128,179 Penn. . . . Dec. 12, 1787 5,258,014 Ind. T. .. June 30, 1834 180,182 North . Atlantic divi- South Central divi- sion . sion . Del Dec. 7, 1787 168,493 Mont. . . . Nov. 8, 1889 132,159 Md April 28. 1788 1,042,390 Wy Col July 10. 1890 60,705 D. C Mar. 30, 1791 230.392 Aug. 1, 1870 412,198 Va June 25, 1788 1.656,980 N. M Sept. 9, 1850 153.593 W. Va. . . June 20, 1863 762,794 Ariz Feb. 24, 1863 59.620 N. C Nov. 21, 1789 1,617,947 Utah Jan. \ 1896 207,905 S. C Mar. 23. 1788 1,151,149 Nev Oct. 31, 1864 45.761 Ga Jan. 2. 1788 1.837,353 Idaho July 3. 1890 84,885 Fla Mar. 3, 1845 391.422 Wash. . . . Nov. 11. 1889 349,390 Ore Cal Feb. 14. 1859 Sept. 9, 1850 313.767 1,208,130 South J sion . Atlantic divi- Western division Ohio Feb. 19, 1803 3,672,316 Ind Dec. 11. 1816 2.192.404 U. S. without out- Ill Dec. 3, 1818 3.826,351 lying territory. . . • . Mich.. .. Wis Jan. 16, 1837 May 29, 1848 2.093,880 1,686,880 Minn. . . . May 11.1858 1.301,826 Alaska . . July 27. 1868 32,052 Iowa Dec. 28, 1846 1,911,896 Hawaii. . April 30, 1900 89,990 Mo Aug. 10, 1821 2,679,184 Phil. Is. . N. D Nov. ^, 1889 182,719 Tutuila.. S. D Nov. 2. 1889 328.808 Guam . . . Neb Mar. 1, 1867 1,058,910 Porto ) Rico J Kan Jan. 29, 1861 1,427,096 North i Central divi- Total U. S. with out- sion . Ivine: fArrifnrv 'J *"6 ' 40 RATIONAL ORAltMAft SCHOOL ARltMMEtlC United States Export Trade Country United KiDgdom . . (rerniany Canada Netherlands Mexico Italy British Australasia British Africa Japan Brazil Argentina Russia Total 1901 $598,766,799 184.678.723 107,496.522 85,643.804 36,771,568 84.046,201 30,569,814 24,994.766 21.162,477 11,136.101 11,117.521 6,504,867 1891 9482,295, 796 90,826,332 41.686,882 81,261,766 15,371,870 14,447,004 13,564,931 3.511,668 3.839,384 15.064,346 1,909,788 6,400.857 Ten-Ybab Difference Values op Manufactured and Industrial Products Manufactured Articles 19(tt 1897 Difference Agricultural implements Books mans, etc 82,075,609 988,195 913,513 198,438 385,086 1,634,642 98,476 735,165 182,710 894,330 « 248,466 470,358 80.065 31,583 1,499,769 988,661 839,563 877.549 69.756 1,222,708 Carria&res and cars Copper ingots Cotton cloths Cotton manufactures, other Cycles, and parts of Builders' hardware Sewincr machines Other machinery Total Other Articles 190S 1897 Difference Corn »1.468,390 3,769,577 638,361 5,473,177 4.509,205 1,345,260 667,164 261,688 240,978 557,827 218,995 »1, 770.581 2,548,778 2,415.519 6,987,856 2,626,679 566.584 195,584 47,069 208.195 365,419 188,116 Wheat Wheat flour Coal Cotton Fruits and nuts Furs and fur skins Cotton-seed oil Beef salted or oickled Bacon Hams Total SUBTRACTION 41 3. The columns on the opposite page show the values of the different kinds of goods exported by the United States to Canada during the nine months ending March, 1902, and March, 1897, respectively. Make a list of diflPerences, marking the difference + whenever it denotes an increase, and - when it denotes a decrease. §29. Nnmberg for Individual Work. School Statistics for the Thirty Largest Cities of United States City- Popula- tion. Census 1900 POPUIiA- TION. Census 1890 School Enroll- ment. 1900 NUMBKK TEACH- ERS. 1900 School EXPENDI- TUKKS. 1900 New York 3,437,202 1,698,575 1,293,697 575,236 560,892 508,957 381,768 352,387 342,782 325,902 321,616 287.104 285,704 285,315 246,070 218,196 206,433 204,731 202,718 175,597 169.164 163,752 163,065 162,608 133,859 131,822 129,896 125,560 118.421 108,374 2,492,591 1,099,850 1,046,964 451,770 448,477 434,439 261,353 255.664 298,997 296,908 238,617 242,039 205,876 204,468 181,830 188,932 163,003 161,129 164,738 132,146 105,436 132.716 133.156 133,896 106,713 81,434 105,287 88.150 84.655 88,143 559,218 262,738 151,455 82,712 91,796 65,600 59,635 56,000 48,517 44,285 50,000 31,547 40,303 87,000 41,870 40.069 32,174 27.626 38,591 23.485 27.334 28.280 26,000 24.896 27,181 21.467 20.104 18,855 19,600 21,090 12,212 5,951 3,591 1,751 2,018 1,600 1,303 1,300 1,017 993 1,000 782 ' 966 900 851 1,043 686 650 892 682 650 700 610 692 530 455 377 502 574 485 $21,040,810 7.929,496 4,677,860 1.526.140 3,664,298 1,279,936 1,933.965 1,408,000 1,152,631 1,064.047 1,757.381 455.073 1,251.825 733,510 1,218,660 634.153 555,811 841,000 682,000 729,106 524,065 672,350 682,018 750.180 471.314 835.634 771,132 529.937 409.073 Chicago Philadelphia St. Louis Boston r Baltimore Cleveland Buffalo San Francisco Cincinnati Pittsburg New Orleans Detroit Milwaukee Newark Washington Jersey City ....... . Louisville Minneapolis Providence Indianapolis Kansas City, Mo. . . St. Paul Rochester Denver Toledo Alleerhenv Columbus Worcester Syracuse ... Total Problems like the following may be made from the table, and assigned to different pupils. 42 RATIONAL GRAMMAR SCHOOL ARITHMETIC 1. Find the totals of all columns of the table for the cities whose inhabitants numbered over 500,000 at the 1900 census. 2. Find the totals for the cities whose populations in 1900 were between 250,000 and 500,000; for cities with populations between 100,000 and 250,000. Note. — These intervals may be shortened or lengthened at will. Indi- vidual pupils majr do different parts of the work, thus obtaining a large number of individual problems. In each case the pupil should tell what his total means. 3. Find the increase in population from 1890-1900 of Cleve- land, Ohio; of other cities. 4. IIow many more teachers in 1900 were there in Minneapolis than in Louisville? in Xew York City than in Chicago? in Boston than in St. Louis? in Chicago than in Boston? 5. IIow many more pupils in 1900 were enrolled in the schools of Boston than in those of St. Louis? than in those of Balti- more? of Cleveland? 6. IIow much more money was expended in 1900 on schools in Boston than in St. Louis? than in Cleveland? IIow much more in Cleveland than in Baltimore? than in Buffalo? IIow much more in Pittsburg than in Cincinnati? 7. IIow do the combined school expenditures for Ifew York City, Chicago, Philadelphia, and Boston compare with the com- bined school expenditures of all the rest of the cities in the table? IIow do those of Xew York City and Chicago compare with the total of all the rest? 8. Find the totals for the five columns of the entire table. What is the increase in the total population of these cities from 1890 to 1900? §30* Subtraction of Literal Numbers. 1. How many five-cent pieces are 7 five-cent pieces — 3 five- cent pieces? 2. How many dimes are 12 dimes — 5 dimes? 3. How many 9's are 16 9's-9 9's? 4. How manv c's are I3c - 7c? MULTIPLICATION 43 5. Write the differences : 18a; 26y 43^ 682? 683a 1021«? dx Sij d7z 'd9z 97a 879^ 6. A lot contains 160.T sq. ft., and the house covers 40a; sq. ft.; how many square feet of the lot are not covered by the house? 7. A boy earned 15a; cts. on Friday and spent 10a; cts. on Sat- urday ; how many cents did he save? 8. Tell what number x stands for in these problems: (1) 12 -a; =7. (6) 7a; -5a; = 10. (11) 50a; = 100. (2) 65 -a; = 20. (7) 16a; - 13a; = 15. (12) ^a; = 50. (3) a; - 35 = 60. (8) 9a; - 5a; = 20. (13) fa; = 27. (4) a; - 19 = 7. (9) 7a; = 63. (14) fa; = 7. (5) 3a; - 2a; = 8. (10) 9a; = 108. (15) Ja; = 63. Literal numbers are numbers denoted by letters. MULTIPLICATION §31. Definitions. oral work 1. At 75^ a day, how many dollars does a boy earn in 6 days? This problem may be solved in two ways. We may say he earned the sum of 75^ + 75^ + 75^ + 75^ + 75^ + 75^, which is $4.50. Or, we may say he earned 6 times 75^ (6 x 75^), or $4.50. In either case his earnings are $4.50. 2. Which is the shorter way? How do the addends compare in the first solution? 3. Find in two ways how far a vessel will sail in 4 da., making 22 mi. a day. 4. A factory employee, working by the hour, makes the follow- ing daily record for a week : 8 hr. ; 8 hr. ; 8^ hr. ; 9 hr. ; 9 hr. ; 9^ hr. How many hours does he work during the week? 5. How can you solve this problem in more than one way? Give reason for your answer. 6. When the addends are unequal, as In problem 4, what is the ouly way they can be combined? 7. When the addends are equal, as in the first two problems, in how many ways can they be combined? Which is the shorter 44 RATIONAL GRAMMAR SCHOOL ARITHMETIC way? By what name do we know this shorter way? What then is multiplication? MuUij)lication of whole numbers is a short way of finding the sum of equal addends when the number of addends and one of them are given. The given addend is the multiplicand. The number of equal addends is the multiplier. The result, or sum, is the product. The multiplicand and multiplier atq factors of the product. The sign of multiplication x is read "multiplied by" when it is written after the multiplicand and "times" when it is written before the multiplicand. Thus 22 mi. X = 132 mi. is read "22 mi. multiplied by 6 equals 132 mi.," and 6 x 22 mi. = 132 mi. is read "6 times 22 mi. equals 132 miles." An expression like 6 x 22 mi. = 132 mi. is called an equation. 8. At $3.50 a pair, how much will 12 pairs of shoes cost? $3.50 = cost of 1 pair; $3.50 X 12 = cost of 12 pairs. $3.50 is the multiplicand., 12 is the multiplier .^ and $42.00 is the product. When a problem is expressed in an equation, the equation is called the statement of the problem. WRITTEN WORK Make statements and solve : 1. There are 128 cu. ft. in 1 cd. of wood. How many cubic feet in 15 cords? Statement. 15 x 128 cu. ft. = ? Or, 15 X 128 cu. ft. =x cu. ft. Find the number which should stand in place of x. The number which should stand in place of x in an equation is called the value of x. 2. There are 5280 ft. in a mile. How many feet in 24 miles? 3. 12 lb. of ham at 17^ per lb. = how many dollars? 4. A cold wave from the northwest traveling at the rate of 32 mi. an hour moves how many miles in 24 hours? MULTIPLlCATIOil 45 5. If galvanized telegraph wire weighs 525 lb. per mi. , how many pounds of wire will it take to stretch 6 wires from Chicago to Milwaukee, a distance of 82 miles? 6. How much will this wire cost at 8^ per pound? 7. Find the cost of 275 fence posts at 65^. $. 65 = cost of one post ; 275 = number of posts; $.65 X 275 = 1178.75 (cost of 275 posts) is the equation of the problem. f^.65 In this problem, which number is the _?25 multiplicand? AVhich the multiplier? 325 455 ISO $178.75 This problem may be solved also thus : If each post t2.75 cost 1^, 275 posts would cost $2.75; but as each post ^ costs 65^, the cost of all will be 65 x $2.75, and the {375 work may be arranged as shown. 2550 Or, we may say 275 times $.65 is the same as 65 times $2.75, and multiply as above, $178.75 Since the numerical product of factors is the same, whichever is used as the multiplier, the smaller number is generally taken for the multiplier for convenience, 8. In 64 pk. there are how many quarts? 9. How many pounds in 494 bu. of corn, if in 1 bu. there are 56 lb.? how many pounds in 25 bu.? how many in x bushels? 10. There are 231 cu. in. in 1 gal. How many cubic inches in 587 gal.? in y gal.? in a gallons? §33. Tables. These products must be learned thoroughly : 3X3= » = 3X 3 3X4 = 12 = 4X 8 4 X 3 = 12 = 3X 4 4X4=16 = 4X 4 5 X 3 = 15 = 3X 5 5X4 = 20 = 4X 5 6X3 = 18 = 3X 6 6X4 = 24 = 4X 6 7X3 = 21 = 3X 7 7 X 4 = 28 = 4 X 7 8X3 = 24 = 3X 8 8X4 = 32 = 4X 8 9X3 = 27 = 3X 9 9 X 4 = 36 = 4 X 9 10X3 = 30 = 3X10 ;0X4 = 40 = 4X10 11X8 = 33 = 3X11 11X4 = 44 = 4X 11 12 X 3 = 36 = 3X13 12X4 = 48 = 4X12 4C RATIONAL ORAVMAR SCHOOL ARITHMETIC §34, Learn each of these in the same way: 3X6 = 18 3X7 = 21 3X8 = 34 3X9 = 27 4X0 = 24 4X7 = 28 4X8 = 32 4X9 = 36 5 X 6 = 30 5X7 = 35 5X8 = 40 5X9 = 45 6 X 6 = 36 6X7 = 42 6X8 = 48 6x9 = 54 .7X6 = 42 7 X 7 = 49 7 X 8 = 56 7X9 = 63 8X6 = 48 8 X 7 = 56 8 X 8 = 64 8X9 = 73 9X0 = 54 9 X 7 = 63 9 X 8 = 72 9X9 = 81 10X6 = 60 10 X 7 = 70 10 X 8 = 80 10X9 = 90 11X6 = 66 11X7 = 77 11X8 = 88 11X9 = 99 12 X 6 = 72 12X7 = 84 ORAL 12X8 = 96 WORK 12X9 = 108 1. At current prices, find the cost of the following article^. J lb. Oolong tea. 2. 2 erasers. 3 doz. eggs. 5 lb. ham. 3 lb. leaf lard. 2 lb. Java coffee. 3 lb. best quality of butter. 3 tablets linen paper. 3 packages white enyelopws. 4 doz. pens. 2 lead pencils. 2 bottles writing fluid. 2 doz. oranges. 2 doz. lemons. 1^ doz. bananas. 1 pk. apples. 4. 8 lb. rib roast. 2^ lb. porterhouse steak. 3 lb. lamb chops. 2 lb. sausage. 5. 3 cd. hard wood. 2 T. hard coal. 1^ cd. pine slabs. 4 T. soft coal. 2 T. hay. 3 bu. oats. 2 bu. corn. 4 bales straw. 7. 5 lb. tomatoes. 8 bunches radishes. 3 pk. turnips. 1 pk. beets. 3 qt. Lima beans. 8. 3 pr. shoes. i doz. tennis balls. 3 baseball bats. 2 catcher's gloves. 1 sweater. MULTIPLICATION 47 §36. Farm Account Keeping. — The forms below show how a cer- tain farmer keeps a systematic account of receipts and expendi- tures with each of his fields. The fields referred to are those shown in the drawing of Fig. 4, p. 7. WRITTEN WORK The items for which money was paid out or received were put down with the dates of payment or of receipt in a day-book thus: Day-Book for Forty- Acre Cornfield 1. Paid for 4 da. work removing stalks @ $1.15, Mar. 18. 2. Paid for 9 da. plowing @ $1.15, Apr. 30. 3. Paid for 2 da. harrowing @ $1.00, May 1. 4. Paid for 3 J da. planting corn @ $1.20, May 18. 5. Paid for 6 bu. seed corn @ 90^, May 18. 6. Paid for 2 da. replanting corn @ $1.15, May 30. 7. Paid for 5 da. harrowing corn @ $1.15, May 30. 8. Paid for 7 da. cultivating ^ $1.15, June 15. 9. Paid for 6 da. cultivating @ $1.15, June 30. 10. Paid for 6 da. cultivating @ $1.15. July 15. n. Paid for cutting 240 shocks corn @ 8i<^, Sept. 15. 13. Paid for husking 1392 bu. corn @ 3^, Nov. 20. 13. Paid for shelling 1336 bu. corn @\^y Jan. 10. 14. Sold 1336 bu. corn @ 38f2^, Jan. 10. 15. Received pay for 3 mo. pasture @ $1.50, Feb. 28. 1. These items are here arranged in the form of a receipt and expenditure account. Fill out the vacant columns, and find the totals and the net profit for this forty-acre cornfield: In Account tvith Forty-Acre Cornfield Expenditures Receipts Mar. 18 Apr. 30 May t " 18 •' 18 •• 30 June 15 '• 30 July 15 Sept. 15 Nov. 20 Jan. 10 Removing stalks, 4 da., @ $1.15 Plowing, 9 da., ® $1.15 Harrowing, 2 da., @ $1.00 Planting com, 3% da., @ Seed corn, 6 bu., @ 90^ Replanting, Si da., (g^ $1.15 Hawowlng corn, 5 da., @ $1.15 Cultivating, 7 da., ® $1.15 6 da., @, $1.15 6 da., @ $1.15 Cutting 240 shocks corn (^ 8^0 Husking 1393 bu. corn @. 30 Shelling 1336 bu. com (^ Total Jan. 10 Feb. 28 1336 bu. com @ 380 3 mo. pasture @ $1.50 Total Expenditures to be de- ducted Net profit from 40 acres '* I)eracre Account closed March 1901 48 RATIOKAL GRAMMAR SCHOOL ARITHMETIC 2. From the following list of day-book items for the twenty- acre wheatfield the account below is arranged. Fill out the vacant columns and find the totals, the net profit on the whole field, and the net profit per acre. Day-Book for Twenty-Acre Wheatfield 1. Paid for 4 da. plowing @ $2.50, Oct. 20. 2. Paid for 3 da. harrowing and rolling @ $2.25, Oct. 25. 3. Paid for 25 bu. seed wheat @ $1.25, Oct. 25. 4. Paid for 21 da. drilling wheat @ 12.50, Oct. 30. 5. Paid for cutting 20 acres wheat @ 75^, July 15. 6. Paid for 2 da. shocking wheat @ $1.75, July 15. 7. Paid for threshing 480 bu. wheat @ 6<', Aug. 80. 8. Paid for 3^ da. help in threshing @ $1.50, Aug. 30. 9. Sold 425 bu. wheat @ 68<^, Dec. 22. 10. Received pay for 4 mo. pasture @ $1.50, Mar. 2. In Account with Twenty-Acre Wheatfield EXPENDITURBS RBCBIPTS Oct. 20 " 25 July 15 •• 15 Aug. 30 •• 30 Plowing, 4 da., @ $2.5') Harrowing and rolling, 3 da., @$2.25 ' Seed wheat, 25 bu., % $1.25 Drilling, 2% da., ® $2.50 Cutting 20 acres wheat @ 75^ Shocking wheat, 2 da., @ $1.75 Threshing 480 bu. wheat Help threshing, 3^ da., @ $1.50 Total Dec. 22 Mar. 2 Wheat, 425 bu., ^ %^t Pasture, 4 mo., @ $1.50 Expenditures to be de- ducted Net profit from the field " '* i)er acre 3. Draw up these facts into an account, and find the totals and the net profit, and treat in the same way as above : Day- Book for Thirty- Acre Pasture 1. Bought 120 bu. corn @ 37^, Apr. 23. 2. Sold 5 yearling steers @ $18.00, Apr. 25. 3. Sold 3 yearling heifers @ $15.00, July 6. 4. Sold 2 milk-cows @ $46, July 8. 5. Bought 4 two-yr. old heifers @ $25, July 10. 6. Sold 8 hogs weighing 2064 lb., @ $5 per 100 lb., July 20. 7. 8 head of hogs, worth $12 each, died July 25. 8. Bought 20 pigs @ $2.50, Aug. 10. 9. Sold 3 two-yr. old colts @ $72, Sept. 30. 10. Sold 2 draught horses @ $195, Sept. 30. 11. Sold 9 four-yr. old steers @ $68, Oct. 25. 12. Bought 10 year ling calves© $12, Nov. 1. 13. Sold 3 Jersey milk*o>ows @ $45, Nov. 15. 14. Paid 6 mo. wages for one man @ $80, Dec. 28. 15. Sold 8 head of hogs @ $15, Jan. 10. 16. Bought 12 head of hogs @ $9. 50, Feb. 14. MULTIPLICATIOK 49 4. Fill out this account of the fifty-acre oatfield. Expenditures » Receipts Mar. 30 " 20 " 20 Aug. 18 *• 18 Sept. 15 •» 15 RemoYing old stalks 5 da. (^$1.86 100 bu. seed oats @ 35^ 61 da. work sowing oats @ $1.25 Harvesting 50 acres oats 4| da. labor in harvesting @ $1.75 Threshing 2148 bu. (gj S0 5 da. help threshing @. $1.50 TOTAIi Feb. 15 " 20 1800 bu. oats @ 2210 85 loads straw ^; $2.75 Expenditures to be de- ducted Net profit from 50 acres " '* per acre 5. Treat the meadow account similarly. In Account with Ten- Acre Meadow Expenditures Receipts Nov. 20 Aug. 30 " 80 " 30 10 bu. timothy seed (^ $2.00 Cutting 10 acres hay @ 35e Raking and shocking hay 5da. @ $1.50 Stacking 15 tons hay @ 55^ Dec. 15 12 tons hay ^ $9.50 Expenses to be deducted Net profit from 10 acres ** " per acre 6 . Draw up the following items into an account like the one above; and find totals and profit or loss to the farmer. Day-Book for House and Bam Lot {10 acres). 1. Apr. 15. Bought 6 bu. seed po- tatoes @ $2 2. " 20. Bought 30 pkg. gar- den seeds @ 10^ a " 20. Bought 4 pkg. garden seeds @ 25^ 4. May 15. Paid 5 da. wages @ $1.25 5. " 25. Sold 100 bunches on- ions @ 21 f 6. " 30. Sold 148 bunches cel- ery @ 3^ 7. •• 30. Sold 200 bunches let- tuce @ 2i^ 8. " 30. Sold 240 bunches as- paragus @ Sf 9. June 30. Sold 14 bu. peas @ ¥ per qt 10. '* 30. Sold 28 bu. green beans @ 4^ per qt. 11. *' 30. Sold 300 bunches as- paragus @ 5^ 12. •* 80. Paid 18 da. wages @ 11.25 13. July 31. Sold 10 bu. new pota- toes @ 11 14. July 31. Paid 15 da. wan;es @ $1.50 " 31. Sold 50 heads cabbage @ 8<25 '' 31. Sold20doz. ears sweet corn @ 8,^ per dozen. Aug. 31. Sold 140 lb. tomatoes 15. 16. 17. 18. 19. 20. 31. Sold 10 bu. early apples @75^ 31. Sold 25 heads cabbage @7^ 31. Sold 15 bu. peaches @ $1. 21. ** 31. 10 doz. ears corn @ If 22. ** 31. Paid 20 da. wages @ $1.50 23. Sept. 30. Sold 55 bu. potatoes @ 65 24. •' 30. Sold 20 bu. Lima beans @ $1.25 25. " 30. Paid 20 da. wages© $1.25 26. Dec. 20. Sold 30 bu. apples @$1 27. " 30. Sold 10 bbl. (40 gal. each) cider vinega'' @ 30/ per gal. 60 RATIONAL ORAMMAR SCHOOL ARITHMETIC 7. Make a statement from these items of general expense, not included in any of above accounts, and find the total profit or loss : 1. Apr. 20. Bought 3 sets of har- 7. Aug. 30. Built corn crib, cost ness @ $28 , 1112.00 2. " 25. Paid tax on 160 acres® 8. Sept. 10. Paid for 120 mo. pas- 25^ turing stock @ $1. 50 3. *• 30. Bought 1 wagon @ $60 9. Feb. 28. Paid for 45 mo. pas- 4. • ' 30. Bought 2 plows @ $35 turing stock @ ^1. 25 5. May 10. Bought 3 cultivators @ 10. " 28. Paid 30 da. fertilizing $25 @ $1.25 6. Aug. 20. Paid 84 rods tile ditch- 11. " 28. Paid 8 da. mending ing @ 50,<^ fence @ $1.25 8. Add all the net receipts and all the net expenditures for the separate fields as given in the records of these fields in prob- lems 1 to 7 above. Find the net earnings for the year of the whole farm. §36. Problems. 1. A double eagle weighs 516 grains, and a gold dollar 25.8 grains. Find the difference in weight between a double eagle and 2 gold dollars. 2. AVhat will a peck of peanuts cost at 5^ a pint? 3. At $50 a front ft., what will 50 ft. of city land cost? 4. If milk costs 0^ a qt. and I buy 2 qt. a day, what is my milk bill for April? 5. The average milk yield of a cow was 6 qt. a da. for 120 da. If this milk was all sold @ G^-^ a qt., how much did the owner receive for it? G. The cow's feed cost the owner $2.50 per mo. of 30 da. How much profit did the owner receive from the cow's milk during the 120 days? 7. A cow gives 12 lb. of milk a da. and the butter made from this milk equals ^V ^^ the weight of milk. How many pounds of butter does the milk furnish in 30 days? 8. If butter is selling for 35^ a lb., what is the butter yield of this cow worth in 30 days? 0. G8 qt. of a certain Jersey cow's milk yield 7 lb. of butter. If milk is selling @ 6^ and butter @ 35^, is it more profitable to MULTIPLICATION 51 sell 68 qt. of the milk of this cow, or to make butter of it and sell the butter? How much more profitable is it? 10. The pulse of a healthy man beats 72 times per minute. The heart contracts once for each pulse beat. How many times do the muscles of a healthy man's heart contract in 1 hr.? in 1 da. at the same rate? 11. Count your own pulse beats for 1 min. and find how manj times your heart beats in a day at this rate. 12. Tie a heavy ball to a fine string, or split a bullet and fasten it to a thread, and suspend the string to a hook or nail so that 1^ ft. of it may vibrate freely. Start it swinging and count the number of vibrations per minute. At the same rate how many timeS would the ball vibrate in 24 hours? 13. Solve the same problem with the suspending string 3 ft. long; with the string 3 ft. 3 in. long. 14. My watch ticks 270 times per minute; how many times does it tick in 1 hr.? in 1 day? 15. Light travels 186,600 mi. per second ; how far does it travel in 1 day? 16. A man smokes 3 cigars a day, and pays 15^ apiece for them. How much does this add to his expense account in 365 days? §37. Multiplying by Factors. 1. 7x8 = 56 28x2 = 56 14x4 = 56 7, 8, 28, 2, 14, and 4 are all factors of 56. 2. Write the factors of 44, 48, 96, 35, 84, 81, 49, 108. 3. The two equal factors of 144 are 12 and 12. Give the two equal factors of 4, 9, 25, 49, 36, 121. 4. 8x2 = ? 8x2x5 = ? 8x10 = ? 5. 12x2 = ? 12x2x3 = ? 12x6 = ? 6. 20x4 = ? 20x4x3 = ? 20x12 = ? 7. 25x4 = ? 25x4x6 = ? 20x24 = ? 8. 64x9 = ? 64x9x7 = ? 64x63 = ? 9. Instead of multiplying a number by 15, by what two num- bers in succession may I multiply it and get the same product? 52 SATIOXAL 6HAHMAR SCHOOL ARITHMETIC 10. Multiplying any number by 21 gives the same product as multiplying it by what two numbers in succession? 11. 50 X 2 X 4 gives the same product as 56 multiplied by what single number? 12. Multiplying a number by several factors one after another gives the same product as multiplying it by what single number? A number which has factors other than itself and 1 is called a composite number. A number such as 3, 5, 17, etc., which has no factors other than 1 and itself, is 9, prime number. 13. Instead of multiplying a number by a composite number, in what other way may I multiply it to obtain the same product? 14. Which of these numbers are prime, and which composite : 24, 25, 19, 6, 2, 21, 23, 84, 45, 43, 27, 28? §38. Multiplying by 10, 100, 1000, 10,000. 1. 84x10 = ? 84x100 = ? 84x1000 = ? 84x10,000 = ? 2. 327x10 = ? 327x100 = ? 327x1000 = ? 327x10,000 = ? 3. How may any whole number be quickly multiplied by 10, 100, 1000, or 10,000? §39. Multiplying by a Number Near 10, 100, 1000, 10,000. 1. 746x9 = ? Solution.— This is once 746 less than 10 X 746, or 7460 - 746 = 6714. 2. 9G5 X 99 = ? Ans, 96,500 - 965 = 95,535. 3. 965 X 98 = ? Ans. 96,500 - 1930 = 94,570. 4. 965 X 101 = ? Ans. 96,500 + 965 = 97,465. 5. 965 X 102 = ? 6. 873 X 11 = ? 7. 8473 X 1001 = ? 8. 8473 X 999 = ? §40. Multiplying by 25, 50, 12i, 75, 500, or 250. 1. 65x 25 =? 25 =iof 100. i of 65x100 = 1625. 2. 83x 50 =? 50 =^of 100. ^ of 83x100 = 4150. 3. 638 X 12^ = ? 12i = i of 100. i of 638 x 100 = 7975. 4. 132 X 75 =? 75 = J of 100. J of 132 x 100 = 9900. 5. 640 X 500 = ? 500 = i of 1000. 6. 740x250 =? 250 =i of 1000. MULTIPLICATIOK 53 We see from problem 1 that to multiply by 25, we may multi- ply by 100 and take i of the product. 7. Make a rule, different from the ordinary rule, for multiplying quickly by 50; by 12^; by 250; by 500. 8. Make a rule for multiplying quickly by 33^; by 333 J. This shows the importance of remembering that : i of 100 = 50 J of 1000 = 500 J of 100 = 25 i of 1000 = 250 J of 100 = 75 f of 1000 = 750 i of 100 = 121 i of 1000 = 125 |oflOO = 37J i of 1000 = 375 i of 100 = 621 I of 1000 = 625 i of 100 = BH i of 1000 = 875 ,3^ of 100= 6i Aofl000 = 62i i of 100 = 33J i of 1000 = 333J i of 100 = 661 i of 1000 = 6661 9. Make rules for multiplying quickly by the numbers on the right hand side of these equatioQs. §41. Multiplying When Some Digits of the Multiplier Are Factors of Others. 1. From twice a number, how may you obtain 4 times the same? From 4 times a number, how obtain 8 times the same? 2. Multiply 19,279 by 842. SonjnoN.— 19279 842 38558 = 19279 X 2, the first partial product 771160 = 20 times the first partial product 15423200 = 20 times the second partial product 16232918 3. Observe the relations of the digits of each multiplier below, and shorten the work of multiplication, as above. Prove work : 4,783 X 363 29,847 x 248 12,875x442 86,729x393 §42. Checking Multiplication. Any of the above ways may be used to test the correctness of products obtained in the usual way. l_j_ ; V 5t RATIONAL GRAMMAR SCHOOL ARITHMETIC To check by casting out the nines, cast the nines out of the multiplicand, the multiplier, and the product. Multiply the excesses of the multiplicand and the multiplier. Cast the nines out of this product. If this last excess e,quals the excess of the original product, the work is probably correct. Illustration.— -To check 6848x619 = 4238912, excess in 6848 = 8; excess in 619 = 7; excess in 4238912 = 2; 7x8 = 56, excess in 56 = 2. As excess in 4238912 and in 56 are equal, the work is checked. A very useful check against blunders is to examine the facts of the problem and to decide about what the answer must be, before beginning to solve it. §43. Multiplying by Fractional Nambers. 1. How inauy square yards in a room 4 yd. X Si yards? (Here x is read "by,") Solution. — How many sq. yd. fill a strip 'g 1/ 1^ ' — I — LJ 1 yd. wide extending along the long side? ^**^ ** How many such strips cover the floor? 8| sq. FiGUBK 15 yd. X 4 = a: sq. yd. What is a:? 2. What is the cost of 68^ acres of land @ $80? Solution.~$80x68=$5440, and J of 180 = $40. Then 80X68i = 15480. Observe that $80 X 68} means 80 X 68+ i of 80. 3. Tell what these problems mean and then solve them: $G4xl6i = ? 811b. x25i = ? 24 ft. x8f=? 5280 mi. x 14| = ? 807 yd. x 19f = ? 5^^ sq. yd. x 6 = ? 160 A. X 12| = ? 5i sq. yd. x 5i - ? Note. — Solve the last problem by drawing a square 5i units long and 5) units wide, and dividing it up into square units. The problems just solved show that J x 8 means J of 8, or 3 of the 4 equal parts of 8. 4. Give the meaning of these problems and find the value of the letter in each : ixd=x, fx21 = .c. ix66 = y. iixl08=y. ^x76 = z. §44. Suggestions for Problems. Make and solve problems based on the following facts : 1. A steel rail weighs G4 lb. per yd. of length. The distance from Washington to Baltimore is 42 mi. ; to Philadelphia, 138 mi. ; to New York, 227 mi. ; to Boston, 459 miles. Note— There are 1760 yd. in 1 mile. MULTlPLTrATIOK 00 2. There are double tracks between Washington and each of the cities mentioned. 3. The distance from New York to San Francisco is 32G2 lUiles. 4. The distances in miles from Chicago to 14 railroad centers of the United States are given here: Indianapolis '.184 Rochester ft05 St. Louis 383 Baltimore 801 Cincinnati 298 Washington 820 Cairo 364 New Orleans 912 St. Paul 410 New York 913 Omaha 490 Denver 1028 Buffalo 536 San Francisco 5. Galvanized telegraph wire weighing 572 lb. per mi. is nsed for distances over 400 mi., and for distances under 400 mi., wire weighing 378 lb. per mi. is used. 6. 8 wires run between Chicago and Xew York. 7. Galvanized iron telegraph wire costs 0^ per pound. 8. Sound travels in water at the rate of 4708 ft. per second ; in air 1130 ft. per second. 9. Silver is worth 56^ an oz., 12 oz. to the pound. 10. Hay costs $23 per T. ; oats 42^ per b«. A horse is fed 162 lb. of hay and 10 bu. of oats a month. Bedding costs $2 a month. 11. Standard silver is ^V P^^^ silver and ^V copper. A silver dollar weighs 412.5 grains. The total number of silver dollars coined on this basis was 378,160,769. 12. The number of grains in an ounce of silver is 480. The amount of silver bought under the Sherman Law by the United States government to coin into money was 168,674,682 ounces. 13. Light travels 186,600 mi. per second. It requires 448 sec- onds for light to reach the earth from the sun. 14. It reaches the earth from the moon in 1^^ seconds. 15. An 02. of pure gold is worth $20.67. There are 12 oz. in 1 lb. of gold. 16. An Alaskan miner can take away 200 lb. from the mining district. 56 BATIONAL GRAMMAB SCHOOL ARITHMETIC 17. A cu. ft. of granite weighs 170 lb. A granite step meas- ures i' X 2' X 8'. 18. The distance around the driving wheel of a locomotive engine is 22 ft* In going a certain distance the driver turned 5280 times. 19. In the year 1901, 82,305,924 lb. of tea and ^511,041,459 lb. of coffee wore imported into the United States. Tea was worth 48^ and coffee 2G^ per pound. 20. A prize- winning steer weighed 16.03 cwt. (1 cwt.= 100 lb.) and sold for $9.00 per hundredweight. 21. Another steer of the same lot weighed 1622 lb. and sold for $8.85 per hundredweight. 22. A boy bicyclist rode a miles per da. for h days. 23. Pupils should prepare and solve problems based upon price lists obtained from the grocer and the butcher (see p. 4), or from the market reports of the daily papers. §46. Eainfall. — How long is a cubic inch (Fig. 16)? how wide? how high? What is a cubic foot? A. cubic yard? ^X^\^ <^^*^^ Figure 16 Figure 17 A tin box 3 in. square on the bottom and 9 in. high (Fig. 1?) was used by a school as a rain gauge. A second tin box 1 in. square on the bottom and 9 in. high was used to measure the depth of water in the large box. After the water was poured MULTIPLICATION 57 2 foot from the large box into the small box, the depth of water in the small box was measured with a thin stick and a foot rule. 1. The rain gauge was placed one evening where the rain could fall freely into its open top. During the night it rained and the next morning the water was poured from the gauge into the small box. It filled the small box to a depth of 9 in. How deep did the water fill the large box? How many cubic inches of water were caught in the large box? One cu. in. of water for every sq. in. of surface is what is mQant by 1 in. of rainfall. What is 6 in. of rainfall? 2. How many cu. in. of water are there in a layer 1 in. deep in the large box (Fig. 17)? 2 in. deep? 5 in. deep? 9 in. deep? 3. How many cu. in. of water fell on 1 sq. ft. of the ground during a rainfall of 1 in. (Fig. 18)?' of 2 in.? of 3 in.? of 6 inches? The number of cubic units (cu. in., cu. ft., cu. yd., etc.), a vessel holds, when full, is called its capacity. • 4. What is the capacity of a square- cornered box 3" X 3" X 9"? 5. What was the depth of rainfall during a shower if the large box caught enough water to fill the small box half full? to a depth of 3 in.? of 6 in.? of 1 in.? of 2 in.? of 7 inches? 6. How many cu. in. of water fell on 1 sq. ft. of the ground during a shower giving 1 in. of rainfall? 2 in.? ^ inch? 7. During a rainfall of 1 in. how many sq. ft. of ground received enough water to make 1 cu. ft.? 2 cu. ft.? 12 cubic feet? 8. During June, 1902, the rainfall in the vicinity of Chicago was 6i in. During this month how many cu. in. of water fell on 1 sq. in. of ground? on 1 sq. ft.? on 12 sq. ft.? on 1 sq. yd.? on 30 sq. yd.? on ^ sq. yd.? on 30^ square yards? 9. Just before a shower set an uncovered bucket where the rain may fall freely into it. After the shower measure the depth of the water in the bucket to find the depth of rainfall. s" a" s*' s" Figure 18 58 RATIONAL (JRAMMAR SCHOOL ARITHMETIC Find how many en. in. or cu. ft. of water fell on each sq. ft.? each sq. yd.? each sq. rd. of the ground? Note. — The bucket used must have the same size at the top and bot- tom, with straight sideS. Why? 10. During the first 9 mo. of the year 1902, the region about Chicago received 32 in. of rainfall. IIow many cu. ft. of water fell during this time on a garden bed 10' x 14'? on a garden 48' x 84'? over a city block 250' x 350'? 11. If the walls of your schoolroom were water-tight, how many cu. ft. of water would it hold if it were filled 1 ft. deepT 2 ft. deep? 5 ft. deep? 8 ft. deep? to the ceiling? 12. Find the areas of these rectangles : r/'x8", 12"x28", 40'x64', 376' X 486', 9' x x\ 9 yd. x a;yd., a mi. X }> mi., x units x y units. How can you find the number of square units in any rectangle? Note.— a? X y is written ocy and read *'a:, y.** « 13. How many cubic feet are there in 1 cu. yd.? in 18 cu. yd.? 14. What is the capacity of a square-cornered box 3" x 3" x 9"? 3' X 3' X 9'? of a square-cornered room or space, 3 yd.x 3 yd. X 9 yd.? 3 rd. X 3 rd. x 9 rd.? 3x3x9? 3 x 3 x a? 15. Find the capacity of a square-cornered box 3" x 4" x V \ 3"x4"x8"; 3'x4'x8'; 3'x4'xl5'; 3 yd. x 3 yd. x 3 yd.; 3 units X 4 units x 15 units; 3x4xir; 3xa;xy; a units x J units X c units. Note. — ^The product a? X 2^ X « is written xyz and read "a?, y, «." 16. How can you find the number of cubic units in any square- cornered vessel? 17. Find the total weight of a snow load of 25 lb. per sq. ft. on a flat rectangular roof 25' x 48'. Find the weight of a load of 15 lb. per square foot. 18. Find the total weight of the shingles and sheathings for both sides of the roof shown in Fig. 19, if shingles weigh 3 lb. per sq. ft. , and sheathing 5 lb. per square foot. Note. — The eaves project 2 ft. over the plate. MULTIPLICATION 59 19. Find the weight of the snow load on both Bides of the roof, the weight on each sq. ft. of surface being 12 pounds. 20. Find the cost at 3^ per sq. yd. of lathing and plastering the four walls and the ceiling, no allowances being made. 21. What is the area of the square ABCD of Fig. 19? What is the area of the triangle DE(Tt 22. A strong gale, giving a pressure of 16 lb. per sq. ft., blows squarely against the end of the building. What is the total wind pressure against the end including the gable? 23. How many cu. ft. of space are inclosed by the walls to the base, Z>C7, of the gables? FIGUBB 19 §46. Algebraic Problems. 4 X ic is written ix. If ix = 24, what is the value of a;? Notice that the equation 4a; = 24 is the statement of this prob- lem. A boy reads 24 pages of a book in 4 days ; how many pages does he read a day? What does x stand for in this problem? 1. What number does x stand for in these equations? (l)3a;=18. (4) a;- 8 = 15. (7)a;+ 9 = 17. (10) 5a; + 2a: = 70. (2) 6a; = 48. (e5)a;-18= 4. (8) a; 4- 6 = 22. (11) 9a;- 3a; = 36. (3) 8a; = 72.' (6) a? - 9 = 8. (9) a; + 16 = 25. (12) 8a; + 5a; = 26. 2. Write the sum of a and J. 3. Write the difference of a and h, 4. Write the product of a and })\ oi a and I and c; of a; and y and z, 5. Answer the following questions if a = 9, d = 8 and c = 6 : 4« = ? 12S = ? 9c=? «d = ? 4aJ = ? alc^'t Zdbc^t 6. In the shortest way you can, write eight times a;; seven times y; twenty -five times a times 5; a times x times y. f)0 EATIONAL GRAMMAE SCHOOL ARITHMETIC 7. In the shortest way, write and read a times b\ c times x; fifteen times a times b times x. DIVISION §47. Diyitfion and Subtraction Compared. — oral work 1. A man owes a debt of $12 which he is to pay by work at $2 per day. How much will he owe at the end of the first day? of the second day? of the third day? of the fourth day? the fifth? the sixth? 2. How long will it take to cancel the debt? 3. How many times may 2 be subtracted from 12, leaving no remainder? How many 2's are there in 12? 4. A man buys a horse for $90 and is to pay $15 a month until the horse is paid for. How much does the man owe after the first payment? after the second? the third? fourth? fifth? sixth? 5. How many months will it take to pay for the horse? 6. How many times may 15 be subtracted from 90, leaving no remainder? How many 15's in 90? 7. What is one of the 6 equal parts of 12? of 90? 8. A rectangular plot of ground 8 yd. wide by 18 yd. long is covered with bluegrass sod. How many square yards of sod does it contain? After a strip of sod 1 yd. wide, extending the length of the plot, has been removed, how many square yards of sod remain? 9. How many square yards remain after the removal of 2 such strips? of 3? of 4? of 5? of 6? of 7? of 8? 10. How many times may 18 sq. yd. be subtracted from 144 sq. yd., leaving no remainder? How many 18's are there in 144? What is one of the 8 equal parts of 144? written work 1. There are 160 bricks in a pile; find by subtraction how many loads of 20 bricks each there are in the pile. How mauv 20's are there in 160? DIVISIOK 61 2. Find by successive subtraction how many SB's there are in 616. What is one of the 7 equal parts of 616? 3. Find by subtraction how many months of 30 da. there are in 270 da. What is one of the 9 equal parts of 270? . 4. What number may be subtracted 4 times in succession from 120, leaving no remainder? 5. Find by subtraction how many times $1263 is contained in $5052. 6. Tell bow to find by subtraction how many times one num- ber is contained in another. 7. Find in a shorter way how many 12's there are in 156. 8. Find in a shorter way than by subtraction one of the 14 equal parts of 224. By what name do you know this short way? 9. In the same way find one of the 15 equal parts of 225 ; of 210; of 240. 10. A lawn-mower cuts a strip 18 in. wide. How many strips must be cut to mow a lawn 18 ft. wide? 20 ft. wide? 30 ft. wide? Division is a short way of subtracting one number from another a certain number of times in succession. §48. Diyision and Multiplication Compared. — oral work 1. A horse traveled 72 mi. in 8 hr. ; find the number of miles traveled per hour. 2. 4 bu. potatoes cost $2.40. What was the price per bushel? 3. A room 4 yd. wide contains 24 sq. yd. ; what is the length of one side? 4. 15 lb. sugar cost 90^; what is the price per pound? 5. At 1^6 per ton, how many tons of coal can be bought for $180? 6. A floor containing 132 sq. ft. is 11 ft. wide; what is the length? 7. Find the cost of 10 pk. apples @ 14^ per peck. If 10 pk. apples cost $1.40, what was the price per peck? 8. How many pecks of apples @ 14^ can be bought for $1.40? 9. A train runs 420 mi. in 12 hr. ; find the average number of miles per hour. 62 BATIOKAL GRAMMAR SCHOOL ARITHMETIC 10. I paid 96^ for veal at 12^ per pound; how many pounds did I buy? 11. What is the cost of 3 T. hay at $12 per ton? 12. At $12 per ton, how many tons of hay can be bought for $36? 13. A man paid $36 for 3 T. hay; what was the price per ton? 14. Find the cost of 25 bbl. (barrels) flour at $5. 15. Paid $125 for 25 bbl. flour; what was the price per barrel? 16. Paid $125 for flour at $5; how many barrels were bought? 17. The two factors of a number are 13 and 7; what is the number? 18. The product of two numbers is 91 and one of the numbers is 13; what is the other number? 19. The product of two numbers is 240 and one of the num- bers is 20 ; what is the other? 20. 20 X a: = 240 ; what number does x stand for? 21. Tell what number the letter stands for in each of these equations: 12^ = 96; 8a; = 56; 7a = 56; 9J = 72; 10« = 150; 24m = 240. Division is a way of finding one of two numbers when their product and the other number are given. The product is called the diiridend. The given number is the divisor. The required number is the quotie7it. Illustration.— The product of two factors is 490, and one of the fac- tors is 7. What is the other factor? 70 7)490 "We may also say that the dividend is the number to be divided, the divisor is the number by which the 4ividend is measured or divided, and the quotient is the measure. The sign h- of division is read "divided by," as 60 -*• 12 = 5. 60 -i- 12= 5 may also be written in the following ways: 5. 12)60(5 00^^ , 12)60 ^ 12 ^ 'The first and second are in common use in division. In the third, the line placed between two numbers shows that the number above it is to be divided by the number below it; as DIVISION 63 in ^, 1 is the dividend and 3 the divisor. The fraction itself is the quotient. The line between the two numbers is the division sign. The fourth sign is called the solidus {sol'i-dus). §49. Short Division. 1. How many gallons are there in 296 pints? Solution. — As there are 8 pt. in 1 gal. there are as many 37 gallons in 296 pt. as there are 8's in 296. g )296 2. How many days in 120 hours? 3. At $8 per ton, how many tons of coal can be bought for $240? 4. 984 marbles are distributed equally among a certain number of boys. Each boy has 82 marbles. There are how many boys? 5. Selling at 6 for a cent, how much will a dealer receive for 540 marbles? 6. A man paid $2.60 for 4 bbl. of lime. What did each bbl. cost? 7. I bought 6 lb. of butter for $1.62. What was the price per pound? 8. A dressmaker used 84 yd. of cloth for 6 dresses, allowing the same amount for each dress. How many yards in each? 9. A train ran 150 mi. in 6 hr. Not allowing for stops, what was the average number of miles per hour? Find the value of z in problems 10 and 11: 10. 48 qt. = X gallons. 11. At 6^ I can buy x lb. of sugar for $1.50. 12. 9 papers of needles cost 72^. One paper costs how many cents? 13. In 1901, the number of trains entering Chicago every 24 hr. was about 1320; what was the average number per hour? ORAL WORK What does x stand for in each of the following equations? 1. 63-^7 = .'C 4. 45^3 = a; 7. 120-^3 = x 2. 630 -s- 9 = re 5. 96 -^ 8 = 2; 8. 108 + 9 = a; 3. 48-*- 4 = a; 6. 960 + 8 = a; 9. 72 -- 6 = a; 64 RATIONAL 0RA3IMAR SCHOOL ARITHMETIC When dividend and divisor are small numbers, the quotient is readily seen. We say it is obtained by inspection. When the divisor is a large number, the quotient is not readily found by inspection. §50. Applications. written work 1. In 61,448 qt. how many pecks are there? SoLXTTiON. — In 51,448 qt. there are as many pecks as there are 8 qt. in 51,448 quarts. 8 is contained in 51,000, 6000 times, with a remainder of 3000, Write 6 in the thousands place in the quotient. 3 thousands = 30 hundreds; 30 hundreds and 4 hundreds = 3400. 8 is contained in 3400, 400 6431 times, with a remainder of 200. Write 4 in the hundreds 8)51448 place in the quotient 200 = 20 tens ; 20 tens and 4 tens = 24 tens. 8 is contained in 24 tens 3 tens times, without a remain- der. Write 3 in the tens place in the quotient. 8 is contained in 8 units 1 unit time. Write 1 in the units place in the quotient. In 51,448 qt. there are 6431 pecks. Check: 6431 X 8 = 51,448. 2. There are 5280 ft. in a mile ; how many yards are there in 1 mi.? in 2 miles? 3. If limestone weighs 160 lb. per cubic foot, how many cubic feet are there in a piece of limestone weighing 6400 pounds? 4. If marble weighs 170 lb. per cubic foot, find the number of cubic feet in a piece of marble weighing 5100 pounds. 5. If sand weighs 120 lb. per cubic foot, find the number of cubic feet in a load of sand weighing 4800 pounds. 6. A train of 12 sleeping cars is 840 ft. long. If the cars are all the same length, how long is each car? 7. A steel rail 30 ft. long weighs 720 lb. ; what is its weight per yard of length? 8. An iron beam 24 ft. long weighs 1080 lb. ; what is the weight of a piece of the beam 1 ft. long? 9. A steel girder (or beam) weighs 1728 lb. Each foot of length weighs 48 lb. How long is it? 10. There were 487,918 foreign immigrants to the United States in the year 1901. What was the average number per month? per day? (30 da. = 1 month.) DIVISION 65 11. During 1900 there were 448,572 immigrants. Find the average number per month. 12. Answer the same question for 1891, 1892, 1893, 1895, and 1897, the numbers for these years being in succession 560,319; 623,084; 502,917; 258,536, and 230,832, 13. In Albany, N. Y., there are 30 mi. of street railway, operated by 600 men. What is the average number of employees per mile? 14. From the data hei-e given answer the same question for these cities : City Miles EMPLOYEES St. Joseph, Mo. . Memphis, Tenn. Oakland, Cal... Hartford, Conn. Worcester, Mass Peoria, 111 35 175 70 490 80 560 33 660 43 473 50 275 15. Find the value of x in each case: (1) 3264+ 6 = a: (2) 9432-^12 = 0; (3) 2247 -^ 7 = a; 16. How can you prove the correctness of your work in divi- sion? 17. Find what x equals in these equations: (4) 89,765 -^ 5 = a; (7) 24,568 f- 8 = a; (5) 78,870 -^ 11 = a; (8) 45,960 -h 12 = a: (6) 75,699-^ 9 = .r (9) 1,241,196-^11= a; (.)?-.o ^ ^ X (5, *f = 49 (7) «f- ^ ^ X («) 7 = 90 («) 640 X 990 X = 20 66 §51. Long Diyision. When dividend and divisor are hoth large numbers, it becomes necessarj to show all the steps of the work. 06 RATIONAL GRAMMAR SCHOOL ARITHMETIC 1. 14,487-^33 = ? 30 400) = 439, quotient 33)14487 13200 = 400 X 33 1287 990 = 30X33 297 297 = 9X33 Shorter Form 439, quotient divisor, 83)14487. dividend 132 128 99 297 297 Solution. —Beginning at the left of the dividend; 88 is not oontained in 1, nor in 14, but it is contained in 14,400, 400 times. Write the 400 above the dividend and subtract 400 X 88 = 18, 200 from the divi- dend, leaving 1287. This remainder must also be divided by 88. 88 is not oontained a whole number of times in 1, nor in 12, but it is contained 30 times in 1280. Write the 80 above the 400 over the dividend and subtract 80 X 88 = 990 from 1287, leaving 297. 88 is contained in 297, 9 times, leaving no remainder. Thus we see 83 is contained in 14,487 400 + 80 + 9 = 489 times. The work may be shortened a little by omitting the zeros and writing numbers in the shorter form below. Check: 489 X 33 = 14,487. division is probably correct. Since this is the given dividend, the When a zero appears in the quotient proceed as follows : 2. 18,722^40 = ? 407 46)T8722 184 322 322 Solution. — Begin as above. 46 is contained in 187, 4 times, with the remainder 8. Bring down the 2 in the dividend, giving 32. 46 is not contained in 82 a whole number of times. Write in tens place in the quotient and brin^ down the next 2 of the dividend, giving 822. 46 is contained in 822, 7 times. Write the 7 in units place in the quotient. Check: 407 X 46 = 18,722, which equals the dividend. Complete these equations and check your work : 3. 3,580 -^ 45= 7. 33,768 h- 72 = 4. 15,552-4-64= 8. 35,096-^82 = 6. 18,144^56= 9. 62,328^-84 = 6. 20,083-^72= 10. 44,928^-96 = DIVISION 07 §52. Exercises. 1. There are 52 wk. in a year. My friend is 1872 wk. old; how many years old is he? 2. There are 160 sq. rd. in 1 A. and a farmer pays 75^ per acre for cutting and binding wheat. How much will it cost to cut and bind the wheat on a field 68 rd. by 80 rods? 3. A farmer paid a man $21.00 to shock his wheat, wages being $1.50 per day. How many days did the man work? 4. In 1880, 25 farm wagons sold for $2250 and in 1900, 15 snch wagons sold for $855. How much less was the average cost of a farm wagon in 1900 than in 1880? 5. In 1880 a Minnesota farmer paid $3900 for 12 twine bind- ers and in 1900, 14 twine binders cost him $1680. How much had the average price of twine binders fallen during tl\ese 20 years? C. A steel rail weighing 72 lb. per yard is 30 ft. long. How many naen are needed to carry it, each man carrying 90 pounds? 7. In 25 da. a man earned $56.25 husking corn at 3^ per bushel. How many bushels per day did he husk? 8. A city lot 175 ft. long, containing 8750 sq. ft., sold for $6000. If the short side fronts the street what was the price per foot of frontage? 9. The force required to draw a street car on a level track is 35 lb. per T. (2000 lb.) of the combined (total) weight of the car and its load. What force is needed to draw a car weighing 5600 lb. when it is loaded with 60 passengers whose average weight is 140 pounds? 10. At the speed of an ordinary horse car the pull of a horse equals about 125 lb. of force in drawing the car. How many horses will be needed to draw the car of problem 9, no horse to draw more than 125 pounds? 11. At a slow walk a horse can exert about 330 lb. of force. The force required to draw a loaded wagon on a level pavement is TfV of the weight of the wagon and load. A coal wagon weighing 4860 lb. is loaded with 4 T. of coal. How many horses will be needed to draw the load over a level pavement, no horse drawing more than 330 pounds? (J8 NATIONAL 6RAMMA11 fiCflOOL ARtTHMETlC 12. A horse can exert 1540 lb. of force for a few minutes. A box car weighing 30 T. is loaded with 32 T. The force needed to move the loaded car is ^ of the combined weight of the car and load. How many horses will be needed to start the car on a level track? §53. Larger Numbers. 1. 5,128,672-1-9272 = ? 653 9272)5128672 46360 49267 46360 29072 27816 1256 Solution. —The divisor, 9272, being too large to use readily, we first use a trial divisor. For the same reason, we select a trial dividend. 92, the trial divisor, is contained in 612, 5 times; but as the whole divisor con- tains four digits the partial dividend must be enlarged. 9272 is contained in 61,286, 5 times. The quotient fi^re 5 is of the same order as the last figure of the trial dividend, wiiich is hundreds. We write 5 in hundreds place in the quotient. Multiplying the whole divisor hj the quotient ngure we have the product 46,860. Subtracting this prod- uct from the trial dividend, 4926 remains. 4926 hundreds = 49,260 tens ; and 49,260 tens + 7 tens = 49,267 tens. Con- tinue in the same manner with each step that follows. The last subtraction gives a remainder of 1256. This remainder must also be divided by the divisor 9272. 5,128,672 -«- 9272 = 553J}f f 9272 553 27816 46360 46860 5127416 1256 5128672 The 553 is the whole, or integral part of the quotient and the mi is the fractional part. Check: Multiply the divisor bv the quotient, and to the product add the remainder. The result should equal the dividend. Solve the following problems and check your work: 2. 131,320-^530= 4. 630,861 + 2731" 3. 195,930 + 624= 5. 1,057,536 + 4352 « §54. Geography. 1. From the table of §23 the area of Massachusetts is seen to be 8315 sq. mi., and that of Illinois is 50,050 sq. mi. How many states the size of Massachusetts could be made from Illinois? DIVISION ^9 2. From the same table the area of Xew England is found to be 66,465 sq. mi. How many states as large as New England could be made from Texas? 3. Make and solve other problems like these, using the table. 4. The same table shows the area of Connecticut to be 4990 sq. mi. and its population for 1900. to be 908,420. How many persons per square mile are there in Connecticut? Note. — In problems such as this, where the fractional part of the Quotient has no meaning, drop the remainder if it is less than half the ivisor, and add one unit to the whole part of the quotient if the remain- der is more than half of the divisor. 5. The table of §28 gives the population of Connecticut for 1890 as 746,258. What was the population of Connecticut per square mile in 1890? . 6. Answer questions 4 and 5 for your own state. 7. From the same table answer questions 4 and 5 for Okla- homa territory. 8. Make and solve similar problems for any states you are studying in your geography. 9. The area of Switzerland is 15,781 sq. mi. and its popula- tion is 2,933,334. What is the population of Switzerland per square mile? 10. 640 A. = 1 sq. mi. How many square miles in 534,528 acres? 11. The area of the state of Texas is 265,780 sq. mi. ; that of New Jersey is 7815 sq. mi. How many states the size of New Jersey could be made from Texas? How many the size of Delaware, Avhich contains 2050 square miles? 12. Porto Kfco has an area of 3531 sq. mi., and a population of 953,243. How many inhabitants does it support to the square mile? 13. The greatest ocean depth found is 31,614 ft. near the island of Guam, in the Pacific ocean. The highest mountain in the world is Mt. Everest in Asia, which rises 29,002 ft. above sea level. Find the difference of level in miles between the greatest ocean depth and the greatest land altitude. 70 RATIONAL GRAMMAR SCHOOL ARITHMETIC 14. The state of New York, with an area of 49,170 sq. mi., supports a population of 7,208,894. How many inhabitants does it average to the square mile? 15. Hawaii has an area of 6449 sq. mi., and a population of 154,001. Find the average population per square mile. 16. In 1891 the total wheat area of North and South Dakota was 4,882,157 A. ; the yield was 81,819,000 bu. What was the average yield per acre? 17. In the year 1900 Kentucky had 22,488 A. of rye under cultivation. The total yield was 292,344 bushels. "What was the average yield per acre? 18. In the year 1890 there were 210,366 persons employed in manufacturing in Chicago. The total wages paid amounted to $123,955,001. What was the average wage paid to each person? 19. In 1880, 3519 factories in Chicago together yielded $249,022,948 worth of products. In 1890, 9977 factories yielded $577,234,446 worth. During which year was the average product per factory greater, and by how much? 20. A comparison of the density of population (population per square mile) may be obtained by finding the density of population of these countries: Country AREA Population Density German Empire Great Britain 208,880 120,979 115,903 204,092 110,646 8,660,395 3,688.110 56,345,014 41,454,578 26,107,304 38,641,333 82,449,754 185,000,000 76,212,168 Austria France Italy Russia United States Note. — Only the whole numbers need be found for these quotients. §55. Diyision by Multiples of 10.— oral work 1. Multiply each of these numbers by 10: 568 1268 306 186.50 $8.65 2. Multiply each of the same numbers by 100 ; by 1000. 3. Divide each of these numbers mentally by 10: 5680 12680 3060 $865.00 $86.50 DIVI8I0K 71 4- Make a rule for dividing any number quickly by 10; by 100; by 1000; by 1 with any number of zeros after it. 5. Name these quotients orally : 60 -^ 10 = ? 6 -^ 2 = ? 60 ^ 20 = ? 860 ^ 10 = ? 86 ^ 2 = ? 860 -k 20 = ? $64.20 H- 10 = ? $6.42-^2 = ? $64.20-^20 = ? Examining your answers to the questions just asked, make a rule for dividing a number quickly by 20 ; by 200 ; by 2000. 6. 180 H- 10 = ? 18^3 = ? 180 + 30 = ? 630 -i- 10 = ? 63 ^ 3 = ? 630 + 30 = V From these answers make a rule for dividing any number quickly by 30; by 300; by 3000. 7. Make a rule for dividing a number quickly by 40; by 400; by 4000; by 50; by 800; by 1200; by 1500. 8. Make a rule for dividing any number quickly by any whole number of tens, as 40, 70, 90, 160; by any whole number of hun- dreds; of thousands. 9. Cutting off zero from the right of a number has what effect ^n the number? cutting off 2 zeros? 3 zeros? 66 -i- 10 = 6«o. or 6.6. 165 -^ 10 = 16/o, orl6.5. 75-^-100=/o"o. or.75. 478 -f- 100 = 4/o8o» or 4.78. 10. Using first 10 and then 100 as a divisor, give and show the quotients of the following : 400 500 2200 3300 460 790 4280 4860 287 439 9647 5732 11. There are 10 pk. in a barrel. How many barrels in 1488 pecks? 12. There are 60 lb. in a bushel of potatoes. How many bushels in 486 pounds? 13. 100 lb.= 1 cwt. How many hundredweight in 825 pounds? 14. 200 lb. pork = 1 bbl. How many barrels in 7624 pounds? 72 BATIOKAL GRAMMAR SCHOOL ARITHMETIC 16. How many minutes in 4260 seconds? Solution.— When there are ciphers at the right of both n^ dividend and divisor, cut off an equal number of ciphers *y from both and divide. 6J*)4260 IG. How many barrels will be needed for 4800 lb. of beef, allowing 200 lb. to the barrel? §56. Other Methods of Shortening Diviiion. 1. Solve these'problems: 5000)25000 500)2500 50)250 6)25 How do the quotients compare? the dividends?^the divisors? 2. Remembering that the dividends stand above the line and the divisors below, solve these problems : 324^^ 108^, 36^^ 12^ 81 • 27 • 9^ • 3 How do the quotients compare? the dividends? the divisors? 3. How do the quotients, the dividends, and the divisors com- pare in these problems : 256^.^ 64^ 16^^ 4^ 128 • 32 8 * 2 4. To divide 324 by 81 what smaller numbers may I use to get the same quotient? How can I obtain these smaller numbers from 324 and 81? Solution.— We see that, as 824 = 27x13 and 81=27x3, we may write 12 X 27 _ 12 _ 8 X 27 ~ 3 ~ This can be indicated thus : 12x3?^ = 4. 3xaT" Removing these factors is called caiicellation. It can often be used to simplify the division of products. 5. Answer the same questions for 256 -^ 128. These problems show that any factor of both dividend and divisor may be dropped from bothand the remaining factors divided. DIVISION 73 6. Solve these problems by cancellation : ... 34x16 ^ 6x8x3 ,, 15x21x846 ,, ^^ 2x16"" ^^2x8x3 * ^^^ 3x 7x846 * 18 x3x4x67 _, ,5)625 ,,. 324 _ , ,^. 85 _ , ^^ 2x4x3x67 • ^^25 ^^89"' ^^^34"* To use cancellation effectively, methods of finding factors of numbers are necessary. §57. Tests of Divisibility. 1. Which of these numbers are exactly divisible (can be exactly divided) by 2 : 12 24 36 23 45 18 37 40 59 61 What are the last digits of the numbers which 2 will divide? Test for the factor 2: If a number ends in 0, 2, 4, 6, or 8, it can be exactly divided by 2. 2. Which of these numbers are exactly divisible by 10: 24 30 45. 50 700 640 83 765 6400 , What is the last digit of the numbers which 10 exactly divides? Test for the factor 10: If a number ends in 0,_10 exactly divides it. 3. Make a rule for testing whether 100 divides a number. 4. Which of these numbers does 5 exactly divide : 16 18 35 25 60 20 28 65 460 675 1260 What are the last digits of the numbers which 5 will divide? Test for the factor 5: If a number ends in or 5, 5 exactly divides it. 5. Which of these numbers does 3 exactly divide: 12 17 24 81 27 93 64 75 126 324 185 Of the numbers which 3 exactly divides, will 3 also exactly divide the sum of the digits? Of the numbers 3 does not exactly divide, is the sum of the digits exactly divisible by 3? Test for the factor 3 : If 3 exactly divides the sum of the digits of a number it divides the number also. 6. Of these numbers what ones does 9 exactly divide: 126 368 453 729 819 639 2358 74 BATIONAL GKAMMAR SCHOOL ARITHMETIC See whether 9 will exactly divide the sum of the digits of the numbers that 9 exactly divides. Test for the factor 9 : If 9 exactly divides the sum of the digits of a number it also exactly divides the number. 7. Which of these numbers does 4 exactly divide: 113 124 368 560 375 486 1204 See whether of the numbers it exactly divides 4 also exactly divides the number indicated by the last two digits. For example, in the third, 368 can be exactly divided by 4 and so also can 68. Test for the factor 4 : If the number denoted by the last two digits of a number can be divided by 4, the entire number can be exactly divided by 4. 8. Which of these numbers are divisible by 25 : 60 175 285 625 1350 1275 8645 8675 8625 8650 25 divides 175 exactly and 25 also exactly divides 75, which is the number denoted by its last two digits. Is this true of all numbers 25 exactly divides? Is it true of any numbers 25 does not exactly divide? Test for the factor 25: If the number denoted by the last 2 digits of any number is divisible by 25 the entire number is divisible by 25. Test for the factor 6: test for both 2 and 3. Test for any composite factor: test singly for ail the factors of the composite factor. 9. Test for divisibility by such numbers as 36, 216, 27, 49, etc., which contain some factor two or more times. 10. Pick out the numbers of this list which are exactly divisible by 2 : 6 81 65 72 86 129 9864 8643 7986 16,835 29,860 11. Pick out those which can be exactly divided by 3; by 9; by 4; by 5; by 10; by 12; by 15. §58. Checking Division. Division may be checked by multiplying the divisor by the quotient and adding the remainder to the product. If the sum equals the dividend, the work is checked. DIVISION 75 Another check is to divide by the factors of the divisor suc- cessively and note whether the final quotient is the same as that given by the complete divisor. To check by casting out the nines, add the excess in the product of the excesses of divisor and quotient to the excess of the remainder. If the excess of this sum equals the excess of the dividend, the division is probably correct. Illustration. — Check the work of problem 1, §53. Cast the 9's out of the dividend, 5,128,672. The excess is i. Cast the 9's out of the divisor, 9272. The excess is 2. Cast the 9's out of the quotient, 553. The excess is 4. Cast the 9's out of the remainder, 1256. The excess is 5. The product of the excesses of divisor and quotient is 8. The excess of 8 is <9 itself. Add this 8 to the excess of the remainder, giving 13. The excess of this 13 is 4 and as this equals the excess 4 of the diyjr dend, the division is probably correct. It is even more important in division than in multiplication to examine a problem carefully before beginning to solve it. Try to foresee about what the answer must be. This often avoids blunders. Illustration. —1. If it requires 480 slates to cover a square (100 sq. ft) of roof surface, how many squares are there in a roof which requires 13,680 slates to cover it? Pupil should at once notice that if it required 500 slates to cover a square, there would be a little more than 13,680 -«- 500, or 136 -4- 5 = 27 squares and he might guess 28 squares. The aotual division of 13,680 by 480 gives 28.5 squares. First form a rough estimate of the answer and then solve these exercises : 2. $238 was paid for flour @ $3.50; how many bbl. were bought? Suggestion. — How many bbl. would there have been if the price had been $7.00 a barrel? 3. In still air a hawk flew 375 miles in 2J hr. What was its speed per hour? 4. In 3.9 hr. a crow flew 97.5 mi.; find the rate of flight per hour. 5. From a certain cow ^ of the milk was butter-fat. The cow gave 12 lb. of milk per day. How many pounds of butter-fat will the milk from this cow yield in 70 days? 76 KATIONAL GRAMMAR SCHOOL ARITHMETIC §59. Town Block and Lots.— Fig. 20 is a drawing, or scale map, of a town block. Scale: 1 in. equals 100 ft. If possible, pupils should measure and draw to scale a block or field in the neighborhood of the schoolhouse, and in all problems use the numbers they obtain from their own measurements in preference to those given in the exercises. > < >- z Ld X B D ^^i^^^:^^^^^^:^^:^}:^, !^,^^ l li^^i^^^^i^^i:^^^:^^^^^ t MARKET ST. —1 I — I — I i in. divided into tenths j^^^.vss.v^^s^s.^v<.^^^^^^ Figure 20 Measure the drawing in Fig. 20, and find how long the block is between the sidewalks; how wide. Note. — Use the scale of tenths given in the figure. Lay a strip of paper having a straight edge beside the marked inch and mark short tines on the strip to indicate the tenths. DIVISION 77 WRITTEN WORK 1. How many square feet in the area of the block mnpo^ 2. At $20 per ft. of frontage on (distance along) Market street, what is lot L worth? 3. Make problems like 2 for other lots on Market street, using price per front foot of lots where you live. 4. At $12.50 per ft. of frontage on Mathews avenue, what is lot H worth? 5. Make similar problems for other lots on Mathews avenue. 6. At $18 per ft. of frontage on Race street, what is lot G worth? Similar problems should be made by the pupil. 7. Make problems for any, or all, of the lots, at the price per front foot where you live. Note. — Each property holder is taxed to provide funds for foundation material, brick, and labor to pave in front of his property to the middle line of the street. This is called an assessment. 8. The streets are to be paved with brick. The cost of exca- vating (digging out) to the proper depth is 30^ per square yard of surface;* find the cost of excavating a strip 1 yd. wide extending from outside edge of sidewalk to middle of Race street. Ans, 13.00. 9. What would be the cost of excavating a similar strip 1 ft. wide? 5 yd. wide? a strip as wide as frontage of lot E? Ans, $1.00; $15.00; $50.00. 10. The cost of foundation material is 36^ per square yard; find the cost of enough such material for the strip 3 ft. x 30 ft. described in problem 8. Ans, $3.60. 11. Find the cost of foundation material for each of the three strips (1 ft. X 30 ft., 15 ft. x 30 ft. , and 50 ft. x 30 ft.) of problem 9? 12. The cost of the brick to be used is $10 per M. (thousand) and 78 bricks are needed to cover 1 sq. yd. What is the cost of the brick needed for the strip of problem 8? for each of the three strips of problem 9? 1st Ans, $7.80. 13. What is the total cost of excavating, of foundation mate- rial, and of brick for the strip of problem 8? for each of the three strips of problem 9 (1 ft. x 30 ft., 15 ft. x 30 ft., and 50 ft. x 30 ft.)? ♦Use prices current in your community whenever they can be obtained, instead of prices given. 78 RATIONAL GRAMMAR SCHOOL ARITHMETIC 14. Find this total for other lots fronting on either Market or Race street. 15. The labor of construction (making) costs 75^ per sq. yd. What is the cost of labor on a strip of the size mentioned in problem 8? Arts. $7.50. 16. What will be the assessment against lot F for paving for each foot of frontage? 17. Make similar problems for other lots, not including the corner lots. 18. Henry and Mathews avenues, which are 30 ft. wide between sidewalks, are to be paved in the same way as Market and Race streets. Rates being the same as above, what will be the total assessment against lot L? (Omit the street crossings.) 19. Make and solve problems like 18 for other lots. 20. The owner builds a house on lot L. The length of the house is 36 ft. and the width is 30 ft. How many square feet of the lot are covered by the house? 21. The southeast corner of the house is located 30 ft. from the south and east lines of the lot. Locate the three other corners. 22. Find the cost, at 45^ per sq. ft., of the concrete walk 3 ft. wide and 76 ft. long on lot L, as shown in the drawing. §60. House and Furnishingpi. — oral work 1. What is meant by a l-brick wall?* a 2-brick wall? 2. If it takes 7 bricks to make 1 sq. ft. of wall surface when laid in a l-brick wall, how many bricks per square foot are needed for a 2-brick wall? a 3-brick wall? a 5- brick wall? 3. From the dotted lines in Fig. 21, can you tell how measurements may be taken on a wall to avoid counting comers twice? t Inside Measure i Outside-Measure FIGUKB21 ^' -^ brick is 2 in. by 4 in. by 8 in. ; how many cubic inches are there in it? "A l-brlck wall is a wall one brick thick, bricks lying on the largest surfaces. DIVISION 79 5. How many square inches in one end? one edge? one of its largest sarfaces? 6. In locating the foundation of the house on lot L (Fig. 20), in what direction would you wish the line of the front foundation wall to run with reference to the Market street line? 7. Point oat some of the square comers on the plans in Pig. 22. 8. Can you point out any corners that are not square? ' WRITTEN WORK The owner of lot L, Avhich fronts on Market street, builds the house whose foundation plans are given in Fig. 22 and whose first and second floor plans are given in Fig. 23. I T 1: N ^^.. ..1; w4-: 1 *— --ioJi * ^---11^ > FOUNDATION Figure 22 1. What will it cost to excavate a rectangle 21' x 36' to a depth of 6 ft., for the foundations and cellar, at 20^ per cubic yard? Think of the foundation as made up of straight walls such as are shown in the second part of Fig. 22. The mark (') means foot or feet, and (") means inch or inches. 2. The foundation is inclosed by 2 side walls, each 36' long, and 2 end walls, each 18^' long. Point out these walls in both parts of Fig. 22. The foundation walls are all 8' high. Find the area of the north surface of the north side wall. Ans. 288 sq. ft. 3. What other wall has an outer surface equal to the surface mentioned in problem 2? 80 RATIONAL GRAMMAR SCHOOL ARITHMETIC 4. Find the area of the outer surface of the east end wall. A?is. 148 sq. ft. 5. What other outside surface equals this one in area? 6. Masons reckon that it takes 14 bricks for each square foot of outer surface to lay a solid 2-brick wall. If all walls are 2-brick walls, and solid, how many bricks will be needed for the north foundation wall? for the east wall? the south? the west? 1st Arts. 4032; 2d Ans. 2072. 7. How many bricks will be needed for all four of the outside walls? 8. There are two inside foundation walls each 18^ ft. long, also one 10| ft. long, and one 11 ft. long. All are 2-brick walls 8 ft, high. If these walls contain no openings, how many bricks will be needed for the inside foundation walls? Afis. 6552. 9. The outside walls contain 8 window openings each 1^ ft. by 3 ft. If masons allow for one-half the area of all openings in computing (finding) the number of bricks, how many bricks should be deducted (subtracted) for the outside walls? Ans, 252. 10. The inside walls contain 4 door openings each 2^ ft. by 8 ft. How many bricks should be deducted for the inside walls? Ans. 560. 11. Find the total number of bricks needed for the foundation walls and their cost at $9 per M.* 12. If hauling costs 75^ per load of 1^ T. and each brick weighs 6 lb., find the cost of hauling the brick for the foundations.! 13. If the mortar costs $1.25 per M. bricks, what will be the cost of the mortar? (Use 18 M.) 14. Four brick piers 2 ft. by 2 ft. and 4 ft. high support the porch columns. How much will the bricks needed for these col- umns cost at $12 per M., counting 22^ bricks for each cubic foot? (Use here the nearest half thousand, i.e. 1^ M.) 15. While the house was building the owner decided to replace weather-boarding by stained shingles on a belt running around the house (which is 30' x 36') and extending to a distance of 9 ft. below the eaves. Each square yard, thus changed, cost $1.75 ♦ In computing the cost of brick use the nearest whole thousand. t When the last load is fractional, the price for a full load is charged. DIVISION 81 extra, no allowance being made for openings. How much doea this change add to the cost of the house? 16. Hardwood floors were decided upon later to take the place of pine floors in the dining-room and front hall. This increased the price by 12^ per sq. ft. How much did this add to the cost of the house, counting the dining-room full 14' 6" wide and making no allowance for fireplace in hall. Ans, $51.54. FIRST FLOOR SECOND FLOOR Figubess 17. A room is said to be well lighted when the floor area is not more than 6 times the area of the window surface which admits light. If all windows as shown in the plan are 2^ ft. wide and 6 ft. high, and \ of the window space is covered by the sash, is the dining-room well lighted? is the kitchen? the hall? 18. Is your schoolroom well lighted? Are the halls of your schoolhouse well lighted? 19. A hall 6 ft. by 8 ft. is lighted by a pane of glass 2^ ft. by 4 ft. in the upper'part of a door. Is the hall well lighted? 20. The assembly room of a church is 40 ft. by 60 ft. and is lighted by 18 windows, each containing 24 panes of glass, 1 ft. by 1^ ft., and by a large front window containing 40 such panes. Is the room well lighted? 21. A room 12 ft. by 15 ft. has 3 windows, 2 containing 2 panes of glass each, tte panes being 1^ ft. by 3 ft., and 1 coi> taining 2 panes of glass 3 ft. by 4^ ft. The room also has a door 2i ft. by 7 ft. which admits light when open. Is the room well lighted if the window blinds are up and the door is open? Is it well lighted if the blinds are up and the door closed? If the blinds are half-way up and the door open? If the blinds are J of the way up and the door closed? 82 RATIONAL GRAMMAR SCHOOL ARITHMETIC §61. Applications of Cancellation. In all the problems of this list indicate your divisions and multiplications, then apply the tests of divisibility, and cancel the factors found in both dividend and divisor. Multiply the uncanceled factors in the dividend together and divide this product by the product of the uncanceled factors of the divisor. Illustration. — A speed of 102 mi. per hour equals how many feet per second? Indicate the work thus: 34 22 102X528^21 mx»6 34X22 748 ,.^. - = — = = — - = 149|. 60X6^. 5 Ans. 149| ft. per second. 1. It took 225,280 lb. of steel rails to lay 1 mi. of single-track railroad. What was the average weight per yard of the rails? 2. Find the number of cubic feet in a ton of 2000 lb. for each of the substances given in the table. 3. The average speed of American express trains is about 35 mi. per hour for long dis- tances. IIow many feet per sec- ond is this? Note.— 60 sec. = 1 min. ; 60 rain. = 1 hour. Solution. — Put work in this form: 7 22 35 X 52813 ^X»9 154 Lb. per Cu. FT. Cu. Ft. PERT. Granite Limestone.. Marble Sandstone . . Slate Cast Iron . . . Steel 170 160 170 140 170 450 480 60X613 3 = -75- = 511. -^^*- '^li ^t. per second. 4. In both England and America, the average speed of express trains for distances from 100 to 250 mi. is about 40 mi. per hour. How many feet per second is this? 5. For long distances the average speed of English express trains is about 43 mi. per hour. How many feet per second is this? 6. In 1893 an express train in the United States ran 1 mi. at the rate of 98 ml. per hour. How many feet per second is this? DIVISION 83 7. A railroads train ran for 1 min. at the speed of 130 mi. per hour. Find the number of feet per second which it traveled. 8. Sound t]»vels in air at the rate of about 750 mi. per hour. How many feet per second is this? 9. Gannon balls have been thrown at a speed of 810 mi. per hour for a few seconds. This is how many feet per second? 10. The moon moves around the earth at a speed of about 1,840,000 mi. in 30 da. How many feet is this per second? Note.— 24 hr. = 1 day. 11. In 365 da. the earth moves around the sun through a distance of about 584,000,000 mi. What is the earth's speed in miles per second? 12. The earth, by turning on its axis, carries a place on its equator about 25,000 mi. in 24 hours. How many feet is it carried per second? 13. A farmer bought 18 bu. of wheat at 75^ and paid the bill with potatoes at 60^ per bushel. How many bushels of potatoes were required? 14. A farmer bought 80 acres of land at $100 per acre and paid for it with land worth $60 per acre. How many acres were required? 15. A man buys 5 acres of city land at $720 per acre and pays for it with 40 acres of farm land. What price per acre does the farm land bring? 16. How many bricks each 2" x 4" x 8" are there in a straight pile 12" X 36" x 20'? In a straight pile 2' x 6' x 12'? 17. How many lots each 25' x 150' can be made from a town block 250' X 300'? §62. Farm Products of the United States in 1900. In these problems use the tests for divisibility (§57) to discover common factors in divisor and dividend, and then remove these factory by cancellation. The numbers that remain in dividend and divisor are then to be divided in the usual way. Omit frac- tions in the quotients. 1. There were 512 coffee farms in the United States in 1900, 84 RATIONAL GRAMMAR SCHOOL ARITHMETIC having a total area of about 70,200 acres. Find the average size of a coffee farm. Suggestion. — Test for the factor 8 in dividend and divisor. 2. The total value of these 512 coffee farms was about $1,933,- 000. What was the average value of a farm? 3. If the total acreage (number of acres) of the coffee farms was about 70,200, and their total value about $1,933,000, what was the average value of an acre of coffee land in the United States? Suggestion. — Test first for the factor 100 and then for 2. 4. There were 441 farms in the United States devoted to rais- ing a certain food plant, having a total acreage of about 18,900. Find the average size of one of these farms. Suggestion. — Test first for 9, then remove the common factor 7 from both dividend and divisor. 5. The total value of these 441 farms was about $562,500. Find the average value of one of these farms. 6. If the total acreage in these farms was 18,900 and the total value $562,500, find the average value of an acre for raising this plant. 7. There were in our country about 5,720 rice farms, having a total acreage of about 108,800 and a total value of about $17,834,- 000. Find the average acreage and the average value of a rice farm, also the average value of an acre for raising rice. 8. About 2,025 nursery farms had a total acreage of about 165,800 and a total value of $19,146,000. What were the average acreage, the average value of a nursery farm and the average value of land per acre for nursery purposes? 0. For sugar farms these numbers were nearly correct : 7,340 farms, of 2,669,000 acres, worth $150,420,000. Solve problems like 7 and 8 for sugar farms. 10. The numbers for farms devoted to raising flowers and plants were about 6,160 farms, of 42,660 acres, worth $52,460,000. Sulve problems like 7 and 8 for flower- and plant-raising farms. DIVISION 85 11. From these numbers, solve like problems for fruit farms : number of farms 82,200; total acreage, 6,150,000; total value, $440,000,000. 12. Further problems may be made on this table, which is a fairly complete list of the farm products of the country. Pboduct Number or Farms ACRES TS FARMS Value of Farms Total Average Total Averag Tobacco 106.000 9,570,000 $ 215,500,000 Vegetables . . . 186.000 10,160,000 547,000,000 Dairy produce . 358,000 43,300,000 1.693,500,000 Cotton 1,072.000 89.600.000 1,108,000,000 Hay and grain 1,320,000 210,260,000 6,380.000,000 Live stock 1,565.000 355,000,000 7,505,300.000 Miscellaneous. . 1,060,000 113,200,000 2,383,700,000 §63. The Thermometer. — The official record of hourly temper- atures for Chicago from 6 p.m. February 27 to G p.m. March 1, 1902, as given in a daily paper, was as follows: Feb. 27; Night Feb. 28; Day Feb. 28 ; Night MA30H 1; Day 6 p. m 7 p. m .38*' .38 .37 .37 .39 .39 .38 .38 .38 .38 .38 .37 .39 .40 6 a. m. . . 7 a. m. . . 8 a. m. . . 9 a. m. . . 10 a. m. . . 11 a. m. . . 12 m Average. . 12 m . . . .40^ ....41 ....41 ....41 ....43 ....44 ....38 .' .* .' .38 6 p. m 7 p. m 8 p. m 9 p. m 10 p. m 11 p. m 12 midnight 1 a. m ' 2 a. m 3 a. m 4 a. m 5 a. m 6 a. m Sum Average ... .39" .38 .37 .37 .37 .37 .36 .36 .35 .34 .34 .34 .34 6 a. m. . . 7 a. m. . . 8 a. m. . . 9 a. m. . . 10 a. m. . . 11 a. m. . . 12 m 1 p. m. . . 2 p. m. . . 3 p. m. . . 4 p. m. . . 5 p. m. . . 6 p. m. . . Sum Average. . ....3r ....34 ....33 ....33 ....32 ....32 . . . .32 ....31 ....31 ....30 ....29 ....28 ....28 8 p. m 9 p. m 10 p. m 11 p. m 12 midnight Average 12 midnight la. m 2 a. m 1 p. m. . . 2 p. m. . . 3 p. m. . . 4 p. m. . . 5 p. m. . . 6 p. m. . . Average . ....35 ....35 ....37 ....39 ....40 ....39 3 a. m 4 a. m 5 a. m 6 a. m Average Note. — The average of temperatures, or numbers, is found by adding them and dividing the sum by the number of temperatures, or numbers, which have been added. To average means to find the average. 86 RATIONAL GRAMMAR SCHOOL ARITHMETIC Feb. 27 from 6 p.m. to 1. Average the temperatures on midnight; from midnight to 6 a.m. 2. What was the average temperature on Feb. 28 from 6 a.m. to 12 m.? from 12 m. to 6 p.m.? 3. What is the difference between the two aver- ages of problem 1? of problem 2? What do these differences mean? 4. What is the difference between the lowest and highest temperatures from 6 p.m. Feb. 27 to 6 p.m. Feb. 28? This is called the range of temperature for the day. 5. The average temperature for Feb. for 30 yr. is 26°. How much does the average temperature for the 24 hr. following 6 p.m. Feb. 27 exceed the thirty- year average? 6. Find the average temperature for the night of Feb. 28; for the day of March 1. 7. Find the difference between these averages. What does this difference show? 8. Make and solve such problems from outdoor thermometer readings at your schoolhouse. 9. What is the reading of the thermometer of Fig. 24? What would the thermometer read if the top of the mercury column stood at "freezing"? at ^'summer heat"? at "blood heat"? 10. Average the thermometer readings taken at your own school for each hour from 9 a.m. to 4 p.m. for a month, and keep in a notebook a record of your readings and averages. 11. Lay off on cross -lined paper the read- ings of problem 10 as the read- ings of the table from 9 a.m. to 4 p.m. of Feb. 28 are laid off in Fig. 25 ; draw a smooth curve, free-hand, through He points. Such a curve gives a picture of temperature changes. mik ^1 a 1 % ■ 1}^ IID- ITD- toa^ KT 'bo- \T. iC 70- eo- S0-: % ♦tJ- i3D- ^ IQ- \^-. iO: 10^ £0: |30-= ♦a- ^1 Figure 24 50' 40" 30' 20' 10' 01 ::n. "1 11 ^^ ■» —J :- 9 10 11 12 1 2 3 4 A.M. A.M. A.M. M. P.M. P.M. P.M. P.M. February 28 Figure 25 10 11 12 I 2 3 4 P.M. P.M. P.M. M, A.M. A.M.A.M.A.M. Febrttarii 28— March l Figure 26 DIVISION 87 12. From a daily paper obtain the hourly readings from 9 p.m. to 4 a.m. and lay them off to scale as the readings of the table from 9 p.m. Feb. 28 to 4 a.m. March 1 are laid off in Fig. 26. Draw the free-hand curve as before and find whether the day or night temperatures are the steadier (more regular) . 13. Average your readings from 9 a.m. to 4 p.m. and draw a straight, horizontal line on your diagram, at a distance above the horizontal zero line equal to the average, as the line av of Fig. 25 is drawn. This is the average line for these 8 hours. EXERCISES FOR PRACTICE Check results in division by multiplying divisor by quotient and adding the remainder if there is one. SHORT AND LONG DIVISION 1. Divide 1260 by 2; by 3; by 6; by 5; by 7; by 8; by 10. 2. Divide 2880 by 3; by 4; by 5; by 6; by 9; by 10; by 12. 3. Divide 3402 by 3; by 7; by 21; by 4; by 12; by 28. 4. Divide 2520 by 4; by 6; by 20; by 6; by 7; by 30; by 8. 5. Divide 33264 by 6; by 7; by 8; by 42; by 56; by 48; by 9. G. Divide 50400 by 10; by 30; by 40; by 50; by 60; by 90. 7. Divide 201600 by 200; by 300; by 400; by 500; by 600; by 700; by 900. 8. Divide 3360 by 7; by 8; by 56. 9. Divide 2880 by 8; by 9; by 72. 10. Divide 57672 by 6; by 2; by 12; by 72. 11. How may you divide a number by 35 by short division? by 56? by 63? by 80? by 72? by 54? by 48? by 84? by 96? LONG DIVISION 12. Divide 360360 by 120; by 210; by 336; by 416; by 1248. 13. Divide 5045040 by 42; by 105; by 315; by 540; by 702; by 862;. by 1404. 14. Divide6387624by 81; by 320; by 576; by 736; by 826. 15. Divide 8736824 by 632; by 82G1; by 42125; by 143216. 16. Divide 30200610 by 700; by 1764; by 1908; by 36821. 17. Divide 23642901 by 21 ; by 86 ; by 364; by 4081 ; by 19881 ; by 98631. 18. Divide 38083584 by 672; by 816; by 2688; by 8064. «8 RATION^AL GRAMMAR SCHOOL ARITHMETIO BILLS AND ACCOTTKTS g64. Exercises. MON. TXJ. Wbd. TH. HOURS FRI. Sat. Wagbs HOURS HOUBS HOURS HOURS HOURS S CTS. AdftniR 8 8 SH 9 su Benson ....,,. 8 8H SH SH S}i 9 Boyd .... 4 8M 6 6 6H^ 7 Olaussen 8 SH S}i 9 8M DenninfiT SH 9 8>i S}i 9 9 *"^**" "o Doan 8 8 8 8 8 8 FiGURR 27 1. Find the number of hours each man worked. 2. Find the number of hours all the men worked. 3. At 30^ an hour, find the daily wages of each man. 4. Find the amount due each man at the close of the week. 5. Find the amount due to the six men at the close of the week. 6. Which man worked the greatest number of hours? the smallest number? 7. Counting 9 hr. to the day, what is the greatest amount any one man could earn in a week? 8. Counting 8 hr. to the day, how much "overtime" did each man work? 9. For how many extra hours did five of the men work, count- ing 8 hours to the day? 10. If double wages were paid for ''overtime," how much did each man earn during the week? How much did all earn? 11. How many hours did the third man lose? §65. ORIGINAL PROBLEMS 1. Adams paid $4 on a doctor's bill, and put aside $2 for rent. 2. Benson has a wife and two children. Average amount for each one in the family. BILLS AND ACCOUNTS 8i) 3. How mucli does Boyd's loss of time reduce his wages for the week? 4. Claussen paid $10 for clothing, and $3.50 for the week's board. 5. Denning saves $3 a week to pay the premium (yearly charge) on a life insurance policy, pays $3.50 a week for board, and 35^ for laundry. §66. Accounts. A man has on hand at the beginning of the month of February, 1902, $250. He receives during the month $200 for salary and $60 for the rent of two houses. During the month he pays $72.50 on an insurance policy, $110 for household expenses, $25.50 for incidentals, and $5 for books and stationery. His account book shows the following entries : Dr, {Receipts) Cash {Expenditures) Cr» 1902 Feb. 1 ** 15 **27 On Hand . To Salary. "Rent .. 28 On Hand 297 00 $250 200 60 510 00 1902 Feb. 1 By Insurance ** Household Exp. ** Incidental " " Books and Stat'y Balance . $72 110 25 5 297 510 00 1. With what items is cash the man's debtor? With what items is cash his creditor? 2. During the month, how much money did the'man receive? How much money did he actually possess between February 1 and 28? 3. How much was spent? What is the difference between receipts and expenditures? How much more was spent than was received as salary? Draw forms and prepare cash accounts for the foUovring: 4. March, 1902. On hand first of the month $1000. March 5, sold land for $1500; received on the 15th, |l50 for rent; and sold a team for $225 on the 25th. Bought real estate on the 8th for $1000; paid $65.25 for repairs on the 15th; taxes on the 16th 90 RATIONAL GRAMMAR SCHOOL ARITHMETIC amounted to $17.50; household expenses for the month amounted to $100, and personal expenses to $50. 5. June, 1902. A boy's receipts and expenditures were as fol- lows: Received 25^ for mowing the lawn on the 1st; 50^ a day for running errands on the 4th and 5th; 25^ for mowing the lawn again on the 15th. On the 17th he paid $1.25 for a bat. The same day he earned 50^ for delivering packages for the grocer. 6. July, 1902. A farmer sold on the 13th two loads of hay at $20 a load; 4 hogs weighing in all 1000 lb. at 6^; 15 lb. butter at 20^. On the 13th he bought 26 lb. granulJ^ted sugar at 6^; 3 lb. coffee at 35^ ; and dry goods to the amount of $15. On the 26th he sold vegetables to the amount of $1.25, and 9 doz. eggs at 10^. The same day he bought clothing to the amount of $10. 7. Find cost of each purchase, also the amount due, on the bill below : E. S. WiCKWIRE, In account with Mannheimer Bros., Dr. 1902 Jan. 4 - 4 •' 12 *' 14 To 1 Rue at S75.50 ** 1 Overcoat *' 35.00 •* 7 yd. Dress Goods ** 2.00 '•3M'' Silk - 1.50 Amoiint'diie Mannheimer Bros. Received payment, Jan. 31, 1902. Draw forms and prepare similar bills: 8. Edward Morris in account with J. W. Bowlby, Dr., St. Paul, Minn., May 30, 1902. 1 doz. collars at 25^ a piece; i doz. pr. cuflfs at 25^ a pair; 2 neckties at 50ji5; 3 shirts at $1.25; 1 hat at $3.50; 1 pr. shoes at $3.50; 1 pr. gloves at $1.25. Paid, May 30. 9. A. H. Simons bought of Yerxa Bros., Chicago, June 6, 1902, 1 sack appleblossom flour at $1.30; 8 lb. ham at 17^; 15 lb. granulated sugar at 6^; i lb. oolong tea at $1.60; 3 lb. Mocha and Java coffee at 40^ ; 1 bunch of celery at 30^. Paid in full the same day. BILLS AKD ACCOUNTS 91 10. Edward Ryan bought of M. J. Doran, February 7/1902, 2 cords of hard wood at $8.75; 3 T. of hard coal at $8.25; 2 cd. of pine slabs at $3.25. 11. May 4, 1902, Alvin Johnson bought of the John Martin Lumber Co., cash payment, 1000 cedar shingles at $3 per M; 1 bbl. of lime at 68^; 1000 ft. of pine flooring at 65^ per M; 3000 laths at $1.50; 15 lb. shingle nails at 7^. Accounts are not always settled in full at the close of the month. When a portion of the amount due is carried over to the next month, this item appears as the first item on the bill, under the title "accounts rendered." 12. Find amount of this. bill: J. FmESTONE, To Allen, Moon & Co. . Dr. 1902 Feb. 1 - 10 " 10 '' 10 - 10 " 15 •' 15 To Acc't rendered 24 85 '• 2 Sacks of Flour at $2.60 " 25 tb. Ham 18 '' 4 doz. Eggs... 22 ** 61b. Butter 25 " 25 lb. Granulated Sugar. . . .06 '* Vegetables 5.00 By Cash 13. A. L. Parker, in account with Marshall Field, lacks $12.75 of paying the entire bill of February, 1902. His March purchases were as follows: March 5, J doz. towels, @ 50^ a piece; 2 table cloths, @ $6.50; 1 doz. napkins, @ $5.50 a doz. ; 2 bed spreads @ $2.50; ) doz. sheets @ 75^ a piece; J doz. pillow cases @ 20^ a piece; 7 yd. dress goods @ $1.50; findings, $5.25. Paid in full March 28. 14. Edward D. Young in account with P. W. Salisbury, wood and coal dealer, has left oyer from a previous account $8.25. On the 3d of January, 1902, he bought 2 T. of hard coal @ $8.25; 1 cd. of hard maple wood, $8.50; 1 cd. of pine slabs, $3.50. Paid the whole amount January 29. 92 RATIONAL GBAMMAB SCHOOL ARITHMETIO 15. On Nov. 15, I purchased 1 can lima beans® 12^; 1 can peas @ 12^; 1 qt. cranberries @ 12^; 4 lb. pork chops @ 12^; 4 J lb. chicken @ 12^ ; 2 lb. butter @ 32^, and paid the account with a $5.00 bill. What change lyas due me? §67. Problems of the Grocery Clerk.— A grocery boy made the following transactions (sales and purchases), including the filling of the orders, and returned the correct change without an error all in one hour. What change did he return to each customer? 1. The first customer bought Celery $0.05 Potatoes $0.15 Eggs 22 Sugar. .10 lb. @ 4^^ and paid the clerk with a $1.00 bill. 2. The second customer bought Salt $0.10 Apples fO.25 Flour 55 Breakfast food 13 Yeast 02 Eice. .3 lb. @ 8^^. Milk 07 and he paid with $1.50. 3. The third customer bought Tomatoes $0.20 Beans. . i . . .4 qt. @ 7^. Soap 25 Tea 2 lb. @ 60^. Butter. .3 lb. @ 27^. Cauliflower. .3 heads @ 18^. Coffee.. 3 lb. @ 33 J j^. and paid with a $5.00 bill. 4. The fourth customer bought Eggs $0.22 Lampwicks $0.10 Bacon 18 Cheese 16 Peas 13 and paid with a $5.00 bill. 5. The fifth customer bought Commeal $0.25 Melon $0.25 Sweet potatoes 27 Pineapple 20 Candy 10 Lard. .3 lb. @ 13i^. Grapes 18 and paid with a $2 bill. BILLS AKD ACCOUNTS 93 6. The sixth customer was a farmer, who sold to the grocer Eggs 6 doz. @ 18^. Tomatoes. . .2 bu. @ 75^. Butter 15 1b. @ 22^. Potatoes. . . .5 bu. @ 35^. and bought of the grocer Sugar 20 1b. @ ^^. Coffee 31b. @ 25^. Cheese 2 lb. @ 18f How much money should the grocer pay the farmer to balance thb account? 7. A grocer's daily sales for 4 weeks are given below. Without rewriting the numbers find the total sales (a) for each week ; (b) for the 4 weeks ; (c) the average daily sales for the month. Wbbkly Totals Sugar corn. .6 doz. @ 10^. Tea 2 1b. @ 55^. Flour 1 sack, $1. 15 Rice 6 lb. @ Sif MON. TUBS. Wkd. THtTB. FBI. Sat. $28.75 $37.25 $35.18 $68.12 $20.13 $86.58 18.62 9.68 21.83 40.28 37.60 75.76 30.18 29.95 23.61 9.61 10.84 68.94 19.27 39<13 28.16 38.19_ 5.63 98.56 §68. Family Expense Account. Foot the following problems as rapidly as you can work accu- rately. 1. The household expenditures (money spent) of a family are here given for each of the first 7 da. of Oct., 1900. Find the daily expenditures and the total expenditure for the week. Oct. 1. Oct. 2. Oct. 3. Oatmeal. $0.11 Laundry $0.48 Bread $0.15 Sweet potatoes .18 Apples 25 Bread 10 Meat .20 Sundries G3 Bread 20 Crackers 15 Meat 18 Peaches 25 Grapes 13 Dried beef. Tea Bread Celery . . . . Soap .10 .30 .20 .05 .25 Total . Total. Baking powder Cheese Macaroni Bananas Honey Peanuts Tomatoes .... Shoe polish . . Carfare 10 Onions .10 Total.... .25 .10 .15 .15 .20 .05 .15 .10 9G SATIOKAL OBAMMAB SCHOOL ABITHMETIO Datb Apr Mat Junb Jult Aug. 1 $ 52.86 $ 53.48 $ 56.75 $ 56.73 2 3.87 2.50 .67 1.28 3 3.68 2.18 1.20 2.10 4 6.36 1.28 .85 3.50 5 1.86 .25 .75 4.05 6 2.15 8.12 3.80 1.10 7 75 3.62 2.37 .05 8 1.95 .67 2.12 .12 9..... , 3.28 4.10 .55 .15 10 2.18 .10 .96 .35 11 4.10 1.01 2.84 3.60 12 2.15 .86 3.69 .10 13 1.00 1.52 2.50 2.18 14 10 .78 .55 .00 15 1.61 2.58 1.86 1.57 16 3.21 3.65 1.68 .76 17 57 .25 2.75 .56 18 3.76 2.51 .99 3.65 19 15 .86 6.24 .05 20 4.25 1.76 3.26 8.29 21 86 2.08 .25 .00 22 95 1.15 4.86 .00 23 87 3.12 .28 1.68 24 1.16 1.28 .83 1.00 25 3.15 3.75 1.96 2.28 26 1.82 .15 1.76 .05 27 2.78 1.67 3.75 6.28 28 60 2.18 1.28 .00 29 3.25 1.01 8.14 3.16 30 96 3.87 1.00 2.15 31 00 .75 .00 1.58 56.10 t .65 2.50 54.10 6.82 4.06 .04 .60 .00 2.76 1.28 .26 .76 1.52 4.26 .68 1.18 2.12 2.60 .39 .00 1.86 3.10 3.24 .69 3.27 1.58 1.68 3.82 .66 .66 1.68 6.63 2.17 .28 .69 .12 3.00 .17 .12 2.16 6.81 2.60 .06 .05 .10 8.16 7.06 .16 .19 .08 2.60 1.20 2.97 2.00 3.60 3.60 .10 .25 2.65 1.68 .00 Monthly I Totals ) Total 6. Find the daily average per month for each of the 6 months. 7. Find the monthly average for 6 months from April to September. 8. Find the monthly average for the whole year. 9. Find by adding horizontally the total expenditure for the first day of these 6 months ; for the second day. BILLS AND ACCOtJNM 9? Figure 2S §69. The Equation. 1. y lb. of sugar are placed on the left scale pan, and a weight of 10 lb. on the right pan balances it (see Fig. 28). How heavy is y? The balance of these two weights is expressed thus: y = 10. (I) This expression is called an equa- tion^ and is read *'y equals 10." 2. If a 4 lb. weight is now added to the right pan, how many additional pounds of sugar must be placed upon the left pan to balance the scales? Write an equation to express the relation between the weights now in the pans. 3. If X lb. on the right balance y lb. on the left, how would you state the fact in an equation? 4. If 20 lb. are added to the x lb. already on the right,-how many pounds must be added on the left to balance the equation? 5. If 35 is added on the right side of equation (I), what change must be made on the left side to balance the equation? Write the equation thus changed. Just as the horizontal position of the scale beam shows that there is a balance of the weights on the pans, so the equality sign, = , shows that there is a balance of value of the numbers between which it stands. It must never be used between two numbers, or combinations of numbers, that are not equal in value. The number on the left of the sign of equality is called the first member^ or the left side of the equation. The number on the right is called the second fnember, or the right side of the equation. 6. Four equal weights and a 5 lb. weight just balance 21 lb, (Fig. 28.). How heavy is one of the equal weights? Solution.— 4a?+ 6 = 21. Subtract 6 from both sides 5 5 4a; = 16. If 4a? = 16 what is x1 What is the answer to the problem? 7. 3y+2=14; findy. 8. eoa; -f 12 = 32 ; find a;. 9. 7rr-f8 = 29;finda;. 10. 9;2;4- 3 = 75; find ^. 11. 15a + 3 = 48; find a. 98 KATIOKAL GRAMMAR fiOHOOL ARITHMETIC CONSTEXTCTIVE OEOMETET §70. Problems with Euler and Compass. . Problems I. to XI. are to be solved with tuIgt and compass. Keep all the pencil points sharp while drawing and work carefully. The simple instrument shown in Fig. 29 is a form of the compass which will do for these problems. The pencil point of the compass will be called the pencil foot, or pen foot. The other point will be called the pin foot. Problem I. — Draw a circle with i inch FiGXjBB 29 . radius. Explanation. —First Step : Place the pin foot on an inch mark of your foot rale and spread the compass feet apart until the pencil foot just reaches to the next half inch mark. Second Step: Without changing the distance between the compass feet, put the pin foot down at some point, as A, Fig. 30, of your paper and, with the pencil foot, draw a curve entirely round the point A. A curve drawn in this manner is called a circle. How far is it from A to any point of the curve? Any part of the whole circle, as the part from C to D, or from D to Bis an arc of the circle. The point A, where the pin foot stood, is the figure ao center. The distance straight across from B, through ^ to C, is a diameter. BC is a diameter. The distance from A straight out to the circle is a radivs. The plural is radii (ra'-di-i). AC, AB and AD are all radii. What part of the diameter equals the radius? EXERCISES 1. Draw circles with these radii: r; 1"; ir; ir; ir; 2". 2. Draw circles with the same center A, and with these radii: r; r; r; 1; U". Circles whose centers are all at the same point are called con- centric circles. Problem II. — Draw (or lay off) a line equal to a given line. CONSTRUCTIVE GEOMETEY 90 Explanation.— Let the given line AB (Fig. 31) have the length a units. With nder and pencil draw any straight line, as CX, longer than a. Place the pin foot on A and spread the v compass feet until the pencil foot just ^ a g reaches to B. Without changing the dis- \j) tance between the feet, put the pm foot on C j J[ C and with the pencil foot draw a short Figuhbsi arc across the line CX as at D. Then CD is the desired line; for if we call its length a?, we have made x=a. EXERCISES 1. Draw lines equal in length to these given lines: a b c 2. Draw lines having these lengths : 1"; If; 2r; 21"; 3^'. Problem III. — Draw a line equal to the sum of two or more given lines. Explanation.— Let the two given lines be a and b. Fig. 82. First step : Draw the indefinite line CX longer than the combined length of a and o, and make CD equal to a as in ? Problem II. /> Second Step: Spread the com- pass feet apart as far as the length V of b. Then put the pin foot on the crossing point (intersection) of the arc and line at D and draw v^nnnv^) auother short arc at E. How long ^'^"""^ '' is DEI How long is CE^ CE is the desired sum. If we call its length 8, we may write this equation : (L) s = a-fb. With ruler and compass how can you find a line equal to the sum of 3 given lines? of 4? of any number of given lines? This is called con- structing the sum of lines. EXERCISES 1. Construct the sum of the lines in each column, denote the constructed line by 5, and write an equation like (I.) for eacli case: cam a den h e p m 100 BATIOKAL GRAMMAR SCHOOL ARITHMETIC 2. Construct these sums: i"+r+r+r; 2"4.j"+{"+}"; j^+sf+r+r- Problem IV. — Draw a line equal to the difference of two given lines. Explanation. — Let the two given lines be a and h. First step: Draw the indefinite line CX longer than the minuend line a. As above, make the minuend CD, equal S to a. b Second Step: Spread "T — . J. the compass points apart, r^ M^ W jg" as before, as far as the — Tf"'^ length of the subtra- Figure 83 hend line h. This is called '* taking b as a radius." Place the pin foot on D, swing the pencil foot back toward C, and draw the arc at E across the line CX. This makes DE how long? What line now equals the difference d between a and &? The equation for this case is: (XL) d = a'K EXERCISE Construct the differences of these pairs of lines, call each dif- ference d^ point it out in the construction and write the equation for each case : m c f CL Problem V. — Draw a line equal to 2, 3, or 4 times a given line. Explanation. — Let a be the given line. We are simply to draw a line equal to the sum of a and a. (See Prob- . a lemin.) ^ „ We may write C r (III.) p = 2a. '■ vi^ How would you con- P struct 3a? 4a? 5a? 7a? FigukeSI SarV 8i/? 6«? Write and read the equation for each construction. EXERCISE Construct three times these lines and give their equations (like III., above): a c d %a 3m Problem VI, — Divide a given line into two equal parts. CONBTKUCTIVB GEOMETRY 101 Definition. — Dividing a line into two equal parts is called bisecting the line. Explanation.— Let the given line be AB, Fig. 86, call its length a. First step: Spread the compass feet apart a little farther than 4 the length of AB, Place the pin foot ^ " £3 first on A and bring the pencil foot, by estimate, somewhere above the middle of the line a, and draw the arc 1. Then carry the pencil foot down, by estimate, below the middle of a and draw the arc 2. These arcs should both be drawn long enough to make sure that arc 1 passes over arc 2, under the middle point. Second step: Now change the pin foot to B, and without chang- ing the distance between the feet, draw arc 3 cutting arc 1 and 4 cutting arc 2. Call the crossing points C and D. Third step: Place the edge of the ruler on C and D and draw the straight line CD, crossing a to I of a. Test by measuring i The equation for this case is A )^ Figure 85 at E. Either AE or BE is equal Test by measuring with the compass. (IV.) g=la, org = ^. The first is read "g equals one-half a," meaning q equals i of a, and the second is read **g equals a divided by 2." Do they, therefore, mean the same thing? EXERCISES 1. Draw a line J" long and bisect it. 2, Bisect lines of these lengths: f'; ir; 3"; 5"; of; 3}". FlGUBB 86 Figure 37 3. Longer lines may be bisected at the blackboard with crayon and string (Fig. 36), or on the ground with a cord and a sharp stake (Fig. 37). 102 RATIONAL GRAMMAR SCHOOL ARITHMETIC Problem VII. — Construct an equilateral (equal sided) tri- angle with each side equal to a given line. Explanation.— Let a Fig. 38 be the given line. J Draw CX longer than a and make CD = a, as in Problem II. With length a between the compass feet, place the pin foot on C and draw arc 1, by estimate, above the middle of CD. Without changing the distance between the compass feet, place the pin foot on D and with the pencil foot cut arc 1 with arc 2. Call B the intersection (crossing FiGUBB 88 point) of arc 1 and arc 2. With the ruler draw a line from C to ^ and another from D to B. Then CDB is the desired equilateral triangle. EXERCISES 1. Construct equilateral triangles having sides of these lengths: bed e 2. Construct equilateral triangles having these sides: 1"; 2"; 3"; 2J"; 4". 3. With crayon and string construct these equilateral triangles on the blackboard : 6"; 1'; U"; 18"; 24". 4. With cord and nail or stake construct these equilateral triangles on the floor or ground: 6'; 10'; 18'; 1 rod; 30'. _ Problem VIII.— Construct an isosceles (i-sds'-see-lees) trian- gle with the base and the two equal sides equal to given lines. Definition. — An isosceles tri- t angle has at least two sides equal. — — Explanation.— Let b, Fig. 39, . S equal the base, and e, one of tne equal sides. Make CD = h (Problem II). With C as center and radius equal to e draw arc 1. With the same radius and with D as center draw arc 2. Connect their intersection E with C and with D. ^ Then CDE is the desired isos- ^ oeles triangle; for we made CD = b, TTTrirBK « CE = e and DE= h, ^'^^"' ^ Definition.— The side, CD, which is not equal to either of the other two sides is the base. C0K8TRUCTIVE OEOMBTRY 103 EXERCISES 1. The figures in Fig. 40 represent idl forms of the triangle. What is a triangle? Figure 40 2. What triangle has all its sides equal? What is an equi- lateral triangle? 3. What triangles have at least 2 sides equal? What is an isosceles triangle? 4. What triangles have no two sides equal? What is a sca- lene (ska-leen') triangle? Problem IX. — Draw a scalene triangle with sides equal to given lines (no two heing equal). Explanation.— On the line CX make CD = a as in Problem I. With C as center and h as radius, draw arc 1. Then with D as center and with c as radius, draw arc 2 across arc 1. Call their inter- section E. With the ruler draw line EC and ED. Then CDE is a scalene triangle Q. _ with the sides equal in length to a, 6 and c. figure 4 1 EXERCISES 1. Draw a scalene triangle having sides of these lengths: ah c 2. Draw a scalene triangle having sides of 3", 4", and 5"; of 1", li", and 2". Problem X. — With the compass draw a three-lobed figure inside of a circle of \" radius. 104 RATIONAL GRAMMAR SCHOOL ARITHMETIC EXPLANATION.- -Draw a circle with V' radius around some point, as O, as a center. Set the pin foot at any point on the circle, as at 1, Fig. 42. and with- out changing the distance between the compass feet draw the arc from A on the circle through O to B on the circle. With same nulius and with pin foot on B, draw a short arc at fS. Put the pin' foot on 2 and with the same radius as before draw the arc BOCy C being on the circle first drawn. Put the pin foot on C and draw a short arc at 3. Then place the pin foot on 3 and draw an arc from C through O to A. Figure 42 EXERCISES 1. Draw a three-lobed figure using a radius of 1". ^2.'^I)raw a three-lobed figure using as radius this line . 3. Making the distance between the compass points J", and using the points 1, 2, 3, 4, 5, and 6 in turn, draw a six-lobed figure like Fig. 43. 4. Color the lobes of your figure with a red lead pencil of with water colors and the spaces a, b^ c, d^ e, and / with a green or yellow pencil or water colors. figure 4s Problem XI. — Draw a regular 6-sided figure (regular hexagon) within a circle of y radius. Explanation. — Draw a circle with center at O and with \'t radius. Starting at any point on the circle as at i, put the pin foot on 1 and draw the short arc B across the circle, keeping the radius J". Then put the pin foot on 2 and draw the arc 3. Then with pin foot on 3 draw arc 4\ and so on around. How many steps do you find reach once round? Now with the ruler and pencil connect 1 with 2, then 2 with 3, 3 with 4, and finally G with 1. The figure made by the 6 straight lines is the regular hexagon desired. Figure 44 MEASUREMENT 105 EXEBGISES 1. Draw regular hexagons within circles of these radii : 1"; 2"; 2. How long is one side of each of the hexagons drawn in exercise 1? How long is the sum of all the bounding lines of each hexagon? pEFiNmoN.— The sum of all the bounding lines of any figure is called the perimeter of the figure. MEASTTKEHEITT §71. Measuring Yalue. Money is the common measure of the value of all articles that are bought and sold. The unit on which United States money is based is the dollar. The dollar sign is $; thus 5 dollars is written $5 or $5.00. This unit is called the U. S. Standard of value. 1. A dollar is worth as much as how many dimes? nickels? quarters? cents? To measure the yalue of one amount of money by another is to find how many times one of the amounts is as large as the other. 2. Measure $1 by 50^; by 25^; by 20^; by 40^; by 75^. 3. Measure $10.50 by 50^; by 75^; by $1.50; by $5.25. 4. A farm is worth $1000 and a city lot $2000. Measure the yalue of the farm by the value of the lot. 5. Measure the value of a $150 horse by the value of a $15 pig. 6. Measure $75 by $5; by $25; by $15; by $150; by $225. 7. Measure the value of 160 A. of land worth $75 per acre by $100; by $500; by $1000; by $8000; by $24,000. 8. Which is worth the more, 80 A. of land @ $75, or 50 A. @ $112? how much more? 9. Measure $400 by the value of an $8 calf; of a $25 colt. 106 RATIONAL GRAMMAR SCHOOL ARITHMETIC §72. Measuring Length and Distance. Answer these problems, first, by estimate, writing your esti- mates in a notebook ; then answer by actual measurement. Finally, compare your estimates with the results of your measurements and tell how much your estimates differ from the measurements. 1. How wide is the page of this book? how long? How wide are the margins? 2. How wide is a pane of glass in your schoolroom window? how long is it? 3. How wide is the top of your desk? how long? How high is the top of your desk from the floor? How high is your seat? 4. How high is the bottom of the blackboard from the floor? How wide is your blackboard? how long is it? 5. How tall are you? How far can you reach by stretching both arms as far apart as possible? 6. How far is it around your waist? How far is it around your chest, just below your arms, when as much of the air as possible is exhaled from your lungs? What is your chest measurement when as much air as possible is drawn into your lungs? 7. The difference between these two chest measures is called your chest expansion. What is your chest expansion? How does your chest expansion compare with the average for the pupils of your room? Note. — ^You may easily increase your chest expansion by a little practice in deep breathing. 8. How many steps wide is your schoolroom? how many steps long? 9. How many feet long is your room? how many feet wide? 10. How many yards long is it? how many yards wide? 11. How many inches long is your step? how many feet long? how many yards long? 12. How many inches long and wide is your schoolroom? 13. How many feet long is your schoolhouse? how many yards long? MEASUREMENT 107 14. How many steps wide and how many steps long is your school yard? how many feet wide and how many feet long? how many yards? 15. How many steps is it from your home to the school- hbuse? how many feet? how many yards? how many rods (1 rd. = 5i yd.)? 16. How far can you walk in 1 min.? How many miles could you walk in 1 hr. at the same rate? 17. How far is it from your home to a neighboring large city? Answer this by using a map and the scale given with it. How long would it take you to walk to that city at the rate of walking in problem 16? 18. How long will it take a train to run from your nearest station to that city at 24 mi. per hour? 19. Using the scale map of your state (see Geography) find the length and the width of your state; of your county; of the U. S. 20. How long and how wide is a two-cent postage stamp? 21. How long is the diameter (distance across) of a copper cent (see Fig. 30)? of a dime? of a nickel? of a quarter dollar? of a half dollar? of a silver dollar? 22. How long is the radius of each of these coins? 23. Wrap a strip of paper, or a string, around each of these coins and find how far it is around each. 24. What kind of unit do you need to measure and express long distances? medium distances? very short distances? Note. — Such a number as three-eighths, or |, of an inch means three units each one-eighth of an inch long. Such a unit is called 9k fractional unit, and such numbers as i, {, {, \\, ^l, |f , are all said to be numbers expressed in fractional units. Name the unit of each of the six frac- tions just given. 25. How long is the distance around {circumference) a bicycle wheel (wrap a string around the wheel)? How long is its diameter? 26. Answer the same questions for a carriage wlieel ; for the bottom of a bottle; of a can; for the head of a barrel; for any other circles you can find. 108 RATIONAL GRAMMAR SCHOOL ARITHMETIC 27. Arrange these measares thus : Objbct Diameter Silver dollar Half dollar Quarter dollar Nickel Dime Cent Bicycle wheel Carriage wheel Bottle Can Barrel head Fill out columns 2 and 3 of a table like this with your measurements and keep them for later use. 9 8 7 6 5 4. :k 5 2 WLMJm 1 ^ ♦ 567 Scale >V 6 9 10 n IS §73. Measuring 8nr£Eu^s. A square unit is a square 1 unit long and 1 unit wide. 1. The side wall of a room is 9 ft. by 12 ft. What will it cost to lath and plaster the wall at 3^ per square yard? 2. Using your own measures of the side and the end walls of your school- room, answer the same question for its walls and ceiling. 3. How many square feet of black- board surface are there in your room? 4. How many square inches are there in a pane of glass in one of your windows? How many square feet of window surface admit light into your room? 5. This should be at least as great as | of the floor surface of your room. Is it? 6. ^ of an inch in the drawing, Fig. 46, stands for one foot in the room. Measure and find the number of square feet of plastered surface in the whole interior of the room, deducting the part covered by the baseboard and the floort Figure 45 MEASUREMEKT 109 Such a representation of the room as that of Fig. 46, show- ing the walls and the ceiling spread out on a flat surface, is called a development of the room. 7. Draw to scale, from your own meas- ures, the development of your own school- room. 8. A schoolroom is 24' square and 15' high. How many sq. yd. of flooring are needed for the room? How many sq. yd. of plastering are needed for the ceiling? 9. One side wall of the room (prob. 8) contains two windows, each 6' X 8' ; an end wall contains a window 6i'x 8' ; and the other side wall has a window and a door, each 4'x 8'. How many sq. yd. of window and door surface in the entire room? 10. The rest of the wall surface is plastered. How many square feet of plastering are there in the room? 11. A barn roof is made up of two rectangles, each 30' x 40'. How many square feet are there in the roof? How many squares of roof- ing? (A square of roofing is a 10-ft. square =100 sq. ft.). 12. A house roof is made up of two rectangles, one 18' x 40' and the other 30' x 40'. How many squares of roofing in the whole roof? How many sq. yd. of roof? Ceiling End Wdl 3ide Wall End Wall Base Board Floor BaS6 BoArd SideWall Scale ^'1 FIGXTBB 46 Figure 47 110 RATIONAL GRAMMAR SCHOOL ARITHMETIC 13. How many acres (1 acre = 160 sq. rd.) are there in a field 40 rd. X 80 rd.? 60 rd. x 80 rd.? 80 rd. x 80 rd.? 80 rd. x 160 rd.? 14. What is the cost at $2.20 a sq. yd. of paving a street GO' wide and f of a mile long? 15. What is the cost at 90^ per sq. ft. of making a cement sidewalk If yd. wide and f of a mile long? 16. How many square feet of glass are there in a window con- taining 4 panes of glass, each 18" x 1 yard? 17. A rug 9' X 12' is placed in a room 14' x 18', with its long edge parallel to the long side of the room and with its center at the middle of the floor. How many square feet of the floor remain uncovered? 18. The streets of a city cut out a row of blocks like that of Fig. 48. The numbers along the sides are the lengths in feet. How many square feet are there in block -4? in J?? in (7? Definitions. — A four-sided figure, like A^ having square corners^ is called a rectangle, A four-sided figure, like G or Hy having two pairs of parallel sides^ whether the comers are square or not, is ^parallelogram. Figure 49 19. What is the area of the rectangle of Fig. 49? MEASUREMENT 111 20. Examine the parallelograms and the rectangles beneath them and find the areas of the parallelograms. 21. If the length of any parallelogram and its distance square across (altitude) are given, how can we find a rectangle with the same area as the parallelogram? 22. Point out in Fig. 48 a rectangle that has the same area as the parallelogram G. 23. The length of a parallelogram is b ft. and its altitude is a ft. ; what is its area? 24. Call P the area, h the length, and a the altitude of a par- allelogram; write an equation to show how you would use h and a ^ to find P, \ 25. The streets, ST and LM, are parallel. Fig. 48. \ j How many square feet are there in block (r? in //? in \ I ^? ini^? 52jI 26. Study Fig. 50 and find how many square" feet FIGUBB.50 ^YiQXQ are in D of Fig. 48. 27. How can you find the area of a triangle when you know the length of its base and its altitude (shortest distance to the base from the opposite corner)? 28. In the triangle, ABC (Fig. >^- yD 51), call the area T, the altitude y/1 I \v y^ (OB) a, the base (AB), b. Write an ^ X I ^ ^ ^ equation to find T from a and b, ^ 29. Find the area in square feet x figubb 5i of the following triangles of Fig. 48: Z?; C; D\ E\ F, 30. The part of a roof in the shape of a parallelogram has a length (bane) of 18 ft. and a breadth {altitude) of 12 ft. What is the area of the parallelogram? 31. A city lot has the shape of a parallelogram of 400 ft. base and 300 ft. altitude. What is its area? 32. A field has the form of a triangle of G8 rd. base and 46 rd. altitude. What is its area? 33. The triangular gable (see Fig. 10, p. 50) of a houso is 24 ft. at the base and is 14 ft. high. What is tho area of the gable? 112 RATIONAL GBAMMAE SCHOOL ARITHMETIC 34. Find the missing part (base, or altitude, or area) of the following parallelograms : Base Altitudb 18 ft. Abba (1) 28 ft. (2) 40 rd (3) (4) (5) (6) (8) §74. Measuring Yolnme (Bulk) and Capacity. ORAL WORK 40 rd. 160 rd. eOJ rd. 88 rd. m rd. p mi. X in. n rd. q mi. 960 sq. rd. 1600 sq. rd. 28160 sq. rd. a; X y sq. m. Figure 52 Figure 53 1. How many cubic inches are there in a 2-inch cube? a 4-in. cube? an 8-in. cube? a 16-in. cube? 2. How many cubic feet are there in a cubic yard? in a 6-ft.* cube? in a room 3 yd. x 3 yd. x 3 yd.? 3. How many cubic inches are there in a 5-in. cube? in a 10-in. cube? 4. How many cubic inches are there in a 9-in. cube? in an 18-in. cube? 5. How many cubic inches are there in a cubic foot? WRITTEN WORK 1. What is the capacity of a square cornered box 3" X 4" X 6" (see Fig. 54)? 3' x 4' x 6'? of a room 3 yd. X 4 yd. X 5 yd.? a yd. x d yd. x c yards? 2. A box-car 8' x 34' can be filled with wheat to a height of 5 ft. When full how many cubic feet of grain does it hold? 3. If a bushel of wheat = | cu. ft., how many bushels does the car hold? 4. Noticing that a cubic foot of grain (not ear corn) is ^ bu., make a rule for finding the number of bushels a wagon-box or a granary will hold when full. Figure 54 HEASUBEHEKT 113 5. A wagon-box is 2' x 4' x 10'. How many bushels of ear-corn will it hold if f eu. ft. = 1 bu. of ear-corn? 6. How many rectangular solids 4" x 3" x 7", will fill a box 16" X 12" X 28 inches? (See Fig. 55.) 7. A box 30" X 24" x 12" will contain how many blocks 6"x 4" x 3" inches? 8. 128 cu. ft. = 1 cd. How many cords in a straight pile of wood 80 ft. X 4 ft. X 4 feet? :figubb55 9. The volume of a rectangular bin is 1500 cu. ft. If it 18 25 ft. long and 6 ft. deep, what is its width? 10. How many bricks 8" x 4 " x 2" are there in a regular pile 218" x 24" x 48" inches? 11. Measure the length, the width, and the height of your schoolroom, and find how many cubic feet of air it contains. 12. How many pounds does the steel T-beam of Fig. 56 weigh if 1 cu. in. of steel weighs 4i ounces? 13. A steel I-beam 24' long has a cross section of the^form and size shown in Fig. 57. How much does it weigh if steel weighs 486 lb. per cubic foot (metal 1" thick)? 14. A water tank is 3' x 6' x 10'. How many cubic feet of water does it hold when full? 15. There are 231 cu. in. in a gallon; how many gallons does the tank of problem 14 hold? 16. How many cubic inches of grain does a box hold if it is 8 in. square and 4| in. deep? how many gallons? 17. How many cubic inches does a box hold if it is 16 in. square and 8^ in. deep? how many bushels (2150.2 cu. in. = 1 bushel)? 18. How many cubic inches does a box hold if it is 9" X 10" X 12"? about what part of a bushel? 19. Answer similar questions for a box 10 in. square and 10} in. deep; for a box 10" x 12" x 18". Figure 57 114 EATIOKAL GRAMMAR SCHOOL ARITHMETIC §75. Measuring Weight. Beview the oral work of §74. 1. A 3-in. cube of soil weighs 20 ounces; how many pounds does 1 cu. ft. of the same soil weigh? How many pounds does a cubic yard weigh? 2. A 4-in. cube of sand weighs 48 ounces ; how many pounds does 1 cu. ft. of the same sand weigh? How many pounds does a cubic yard weigh? 3. A cubic foot of water weighs 1000 ounces ; how much does a 6-in. cube of water weigh? a 3-in. cube? 4. A cubic foot of iron weighs 480 pounds; how many ounces does 1 cu. in. weigh? Suggestion. — Notice that 1 cu. ft. = 12 x 12 x 12 cu. in., then use cancellation, thus : 10 ^0 4 ^^0 X 10 10x4 40 ,. . , • I, .4 lUlUlt = T73 = "9 ^ ^♦^ ^^^- ^ ^"- ^^- ^'^^^^ ^* ^^• 3 3 5. A cubic foot of oak wood weighs 64 lb. ; how many ounces does 1 cu. in. of oak wood weigh? How many cubic inches of oak weigh just 1 pound? G. A cubic foot of copper weighs 552 lb. ; how many ounces does 1 cu. in. of copper weigh? How many cubic inches of copper weigh a pound? 7. A cubic foot of gold weighs 1200 lb. ; how many ounces does 1 cu. in. of gold weigh? How many cubic inches of gold weigh a pound? 8. A cubic foot of zinc weighs 43G lb. ; how many ounces does 1 cu. in. of zinc weigh? How many cubic inches of zinc weigh a pound? 9. A bushel of coal weighs 80 lb., a short ton weighs 2000 lb., and a long ton weighs 2240 lb. How many bushels of coal are there in a car load of 36 short tons? of 36 long tons? MEA8UREMEKT 115 10. Find the cost of 31 lb. of wire nails @ 4 cents. 11. What is the cost of 1 lb. of sugar selling 22 lb. for a dollar? 12. Postage on first-class mail matter is 2^ an ounce. What would the postage be on a 2f lb. package of first-class matter (16 oz. = 1 pound)? ^13. Find the cost of 2f lb. butter @ 32 cents. 14. New York merchants bought in 1 da. the following: C324 lb. butter at an average price of 22^ cents ; 1988 lb. cheese at an average price of 14J cents; 6840 lb. sugar at an average price of 4} cents; 2780 lb. sugar at an average price of 4.8 cents. Find the total weight and the total cost. 15. Find the total number of pounds and the total value of these purchases : 650 lb. cut loaf sugar @ $5.74 per hundredweight (cwt.); 825 lb. granulated sugar @ $4.80 per hundredweight; 400 lb. powdered sugar @ $5.24 per hundredweight; 700 lb. confectioner's sugar @ $4.89 per hundredweight; 350 lb. extra white sugar @ $4.78 per hundredweight. 16. Find the weight and total value of this shipment : 5400 lb. plain beeves @ $5.70 per hundredweight; 6375 lb. choice beeves @ $5.80 per hundredweight; 8450 lb. fair beeves @ $4.90 per hundredweight; 8625 lb. medium beeves @ $4.60 per hundredweight; 40,700 lb. veal calves @ $6.50 per hundredweight; 43,000 lb. western steers @ $8.50 per hundredweight; 8450 lb. Texas steers @. $3.70 per hundredweight; 25,200 lb. beef cows @ $2.85 per hundredweight. 17. How many tons did the purchases of problem 14 weigh (2000 lb. = 1 T.)? How many tons in the shipment of problem 16? 18. A troy ounce of pure gold is worth $20.67. How much is a troy pound of pure gold worth (12 troy oz. = 1 troy pound)? 116 BATIONAL GRAMMAR SCHOOL ARITHMETIC 19. An avoirdupois pound equals ly^j of a troy pound. About what is the value of an avoirdupois pound of gold? Note. — Add to the value of a troy pound of gold 31 times j}^ of its value. 20. An avoirdupois pound = 16 avoirdupois oz. ; what is the value of an avoirdupois ounce of gold? 21. What is the value of your weight in gold? (Your weight is given in avoirdupois pounds.) 22. A grain dealer received during January 130 carloads of grain, averaging 25| T. each. Counting 60 lb. to the bushel, how jnany bushels did he receive during the month? §76. Measuring Temperature. On the Fahrenheit thermometer the point where water begins to freeze is marked 32°, and the point where water begins to boil is marked 212°. (See Fig. 24, p. 86.) 1. How many degrees are there between the boiling point and the freezing point? 2. At 11 p.m. on a. certain date a thermometer read 32°. The mercury then fell i° an hour for 4 hr. What was the reading at 3 a.m. the next day? 3. The mercury then fell 1° an hour for 5 hr. What was the reading at 8 a.m.? 4. At 3 a.m. on a certain day the reading was 26®. The mercury fell 2° an hour for 5 hr. What was the reading at 8 a.m.? 5. On a certain date the reading was 10°, and the mercury fell on the average lf° an hour for 7 hr. What was the reading at the end of the 7 hours? 6. It then fell 1^° an hour for 8 hr. What was the reading then? 7. The mercury fell from the reading 12° above zero to 3** below zero. How many degrees did it fall? 8 We might write rea.dmg& above zero thus: A. 12°; A. 6**; A 32°; and readings below zero thus: B. 3°; B. 6°; B. SO*", MEASUREMENT 117 How many degrees does the mercury fall from the first of these readings to the second? (1)A. 8° to A. 3°; (4) A. 5° to B. 7°; (7) B. 2° to B. 11°; (2) A. 8°toB. 3°; (5) A. 2° toB. 12°; (8) B. 7^° to B. 13? ; (3) A. 30° to B. 2°;. (6) A. 4^° to B. 4i°; (9) B. 4° to B. 12i°. 9. How many degrees of rise or fall are there from the first of these readings to the second? If the change is a rise^ mark it R; if 9k f ally mark it F. : (1) B. 7° to B. 2°; (5) A. 9° to B. 9°; (9) A. B^to A. 12° (2) B. 2i° to B. 1° ; (6) A; 2*° to B. 2^° ; (10) A. 3° to B. 30^° (3) B. 2i° to A. 1°; (7) B. 15° to B. 6^°; (11) A. 18°toA.67i° (4) A. 3i° to B. 1° ; (8) B. 15° to A. 6^° ; (12)B. 22^° to A. 67^°. 10. Instead of writing an A. for "above zero" readings, it is customary to use the sign (+). What sign would you then sug- gest for "below zero" readings? Tell whether a change from the first to the second of these readings denotes a rise or Bk fall in each case and by how much? (1) +16° to + 100°; (4) +32° to- 3°; (7) -18° to -34°; (2) + 32° to + 212°; (5) + 16° to - 17°; (8) - 18° to -6^°; (3) + 2° to + 161°; (6) + 8° to -30°; (9)- 6° to - 27^°. 11. The 12 o'clock (noon) readings for 4 successive days were as follows : + 82^° ; + 78J° ; + 61^° ; + 534°. What is the average of these 12 o'clock readings. 12. Find the average of these 9 a.m. readings for 6 da. : + 9° ; + 4°; +5°; +12i°;+14i°; + 9i°. 13. Find the average of these 6 readings: - 4°; - 6°; - 2°; -5°; -13°; -12°. 14. Find the average of these 2 readings: -r 8° and -2^. Show on Fig. 24, p. 86, what point on the thermometer is midway between the readings +8° and - 2°. 15. What point is midway between the readings + 13° and - 5°? between the readings + 4° and - 6°? + 12° and - 12"? -4° and -12°? 118 RATIONAL GRAMMAR SCHOOL ARITHMBTIO §77. Measuring Time. The primary unit used in measuring time is the mean solar day. This day is the average time interval during which the rotation of the earth carries the meridian of a place eastward from the sun back around to the sun again. It is the average length of the interval from noon to the next noon. 1. If the time piece is running correctly, how many times does the short hand (the hour hand) of a watch or clock turn completely around from noon to the next noon? 2. How much time is measured by one com- plete turn of the hour hand? 3. What part of a day of 24 hr. is measured by the rotation (turning) of the hour hand from XII to VI? from XII to III? from XII to I? to FIGURE 58 jj^ ^^ ^^^ 4. What name is given to the time interval in which the hour hand moves from XII to I? 5. For measuring shorter periods the motion of the long hand (the minute hand) is used. What part of a day is measured by the movement of the minute hand from XII around to XII again? from XII to VI? from XII to III? from XII to IX? from XII to I? 6. What name is given to the interval of time required for the long hand to make one complete turn? 7. How many times does the minute hand turn around while the hour hand turns around once? 8. The small and rapidly moving hand covering the VI of the watch face is the second hand. How much time is measured by one whole turn of the second hand? 9. Over how much space does the tip of the minute hand move while the second hand moves around once? 10. The second hand circle is divided into how many equal parts? What name is given to the time interval in which the second hand moves over one of these equal spaces? 11. How many times does the minute hand turn around in one MEASUREMENT 119 day? in a week? in a month of 30 da.? in 365 da. (1 common year)? 12. Answer the same questions for the second hand. 13. There are about 365^ da. in 1 yr. If the hour and the minute hands of a watch start at XII and the second hand at 60 at the beginning of a year, how will the hands point at the instant the year ends, if the watch runs correctly and without stopping? 14. Answer question 13 supposing the length of the year to be 365 da. 5 hr. 48 min. 46 seconds. w J Brown I ■0 3 > tp • o I WHDugan !? f ^ AS Park \ JM5mtth Section of Land— Figure 5» §78. Measuring Land. 1. A section of land is a tract 320 rd., or 1 mi., square. It is usually divided into halves, quarters, eighths and sixteenths, as shown. How many square rods make a section of land? 2. An acre = 160 sq. rd. ; how many acres in a section of land? in a quarter section? 3. A section of land is divided up into farms, as shown in the cut ; how many acres are there in the farm belonging to A. S. Park? to H. S. Barnes? to J. Brown? H. A. Dryer? J. M. Smith? P. S. Mosier? J. S. Doe? 4. Point out the south half of the section; the east half; the Ni; theWi 5. Point out the SE i; the NW i; the SW i; the NE \, 6. Point out the N" J of the SE i; the E ^ of the SE \\ the W i of the NE i; the S i NW i; the W i E^. 7. Point out the SW \ of the SE i; the NE \ SW i; the NWiNWi; theNE i SW i. 8. The farm of E. Miner would be described as the N ^ E ^ E i NE \\ describe the farms of the 8 other owners of this section. 9. How many rods of fence would be required to enclose the section and separate it into farms as shown? 10. How many rods long is a ditch starting from the NE corner of H. A. Dryer's farm, thence running due south to H. S. Barnes's 1^0 RATIONAL GRAMMAR SCHOOL ARITHMETIC N line, thence due W to Barnes's W line, thence S to J. M. Smith's N line, thence due W across Park's farm to his W line? 11. Each small square in Fig. 60 represents a section. The ^ large square represents a townsJiip; how long is a township? how wide? 12. How many square miles in 1 Tp.? how many acres? 13. Point out in Fig. 60 the following: (1) NW i Sec. 33. (2) Si Sec. 17; NE^Sec. 17; EJNEi Sec. 17. (3) NW i Sec. 21; XE i NW i Sec. 21; SE i NW i Sec. 21. 14. How many farms of IGO A. each could be made of a township? 15. How long and bow wide is a square 160-acre farm? 6 5 4 3 2 1 7 e 9 K) II 12 18 -ii 16 19 14 13 10 80 ^ a C3 84 30 ee S6 zt 86 25 31 38 4- 34 35 36 Township— FiGUBB flO ^79. Plotting Observations and Measurements. 1. The hourly temperatures from 6 a.m. to 6 p.nx. Decem- ber 26, 1902, were: 6 a.m. 7 a.m. 8 a.m. 9 a.m. 10 a.m. 11 a.m ISO r/o 10° 10° 12° 13° 12 m. 1 p.m. 2 p.m. 3 p.m. 4 p.m. 5 p.m. 6 p.m. 14° 14° 15° 16° 15° 14° 14° Draw 13 equally spaced vertical parallels one for each hour. Draw a horizontal line as OX across the parallels. Using ^" to repre- sent 1 ° ; measure off on the first vertical tlie distance 01 to represent 7°; on the second vertical the distance 7-2 to represent the second 7°; on the third parallel measure off tlie distance 10°, and so on. Mark the top of each measured vertical distance with a dot. FIGURE 61 Draw lines connecting these dots as in Fig. 61. This is called plotting the readings. This line shows the temperature change during the 12 hrs. MEASUREMENT 121 This plotting is very much aided by the use of cross-lined paper, ruled into small squares. A horizontal side of one of the small squares might be used to represent 1 hr. and a yertical side to represent 1° of temperature. Note.— Pupils in arithmetic should have notebooks containing several pages of squared paper for such work as this. 2. Read the out-door temperatures from hour to hour at your schoolhouse and plot your readings as above. 3. The average hourly temperatures from 6 a.m. to 6 p.m., December 26 to January 2, were: 6 a.m. 7 a.m. 8 a.m. 9 a.m. 10 a.m. 11 a.m. 21.5^ 22.1° 22.2° 22.2° 22.5° 23.2° 12 m. 1 p.m.' 2 p.m. 3 p.m. 4 p.m. 5 pm. 6 p.m. 23.9° 24° 24.5° 25.2° 25° 24.7° 24.3° Plot these readings on squared paper, or by measurement, as above in Figure 61. Plot the tenths by estimate. 4. Does the line for the averages for a week agree in a general way with the line for December 26? What does a comparison of the two lines show? This table gives the heights in feet and weights in pounds of boys and girls of ages given in the first column : HEIGHT VH FEKT Weight in Pounds Agb BOTS GIBL3 BOYS GIBL8 1.6 1.6 7.1 6.4 2 2.6 2.6 25.0 23.5 4 3.0 3.0 31.4 28.7 6 3.4 3.4 38.8 35.3 9 4.0 3.9 50.0 47.1 11 4.4 4.3 59.8 56.6 13 4.7 4.6 75.8 72.7 15 6.1 4.9 96.4 89.0 17 5,4 5.1 116.6 104.4 18 5.4 5.1 127.6 112.6 20 5.5 5.2 132.5 115.3 5. Using squared paper, or making a drawing such as is indi- cated in Fig. 62, plot the numbers in the first column horizontally and^ those in the second column vertically. Draw freehand a 122 RATIONAL GRAMMAH SCHOOL ARITHMETIC lAslesJ FIGUBE 62 smooth line through all the plotted points and obtain the curve for growth in stature of boys. At what age do boys cease growing rapidly in height? 6. Using the numbers of columns 1 and 3 obtain a similar curve for girls. At what age do girls cease growing rapidly in stature? 7. Compare the two curves and note when boys and when girls grow fastest? 8. Using the numbers of columns 1 and 4 draw the curve for growth of boys in weight. 9. Use numbers of columns 1 and 5 similarly for girls. 10. Compare the curves of problems 8 and 9 and note any likenesses or differences in the two curves. What do the curves show about the growth of boys and girls? 11. Twelve different rectangles each 12 in. long have, the widths given in the first line and the areas in the second line. Widths in inches.. 1" 2" 3" 4" 5" G" T' 8" 9" 10" 11" 12" Areas in sq. in.... 12 24 36 48 GO 72 84 96 108 120 132 144 Two lines, OX and OY, are drawn at right ang:les. 12 equally spaced lines are drawn par- allel to OY. The horizontal distances Ol, 02, 03 and so on denote the heights, 1", 2", 3' and so on. On a scale of -J" (or the vertical side of a small square) to 12 sq. in. of area, mark off the lengths la, 2b, 3c, and so on, to denote the numbers in the second line above. This gives the points a, b, c, and so on. Make the construction to the scale indi- figube 63 cated or to some other convenient scale, and place the straight edge of a ruler along the points, such as a, b, c, and so on. On what kind of line do the points seem to lie? 12. The bases of 12 triangles are each 16 in. long and the alti- tudes (heights) and the areas in square inches are: ■ii, rnjjf t^i 'UjIfqW'^ M- ^ ■ B » t M rt 1" 2" 3" 4" 5" 6" 7" 8" 9" 10" 11" 12 8 16 24 32 40 48 56 64 72 80 88 96 MEASUREMENT 123 Using any convenient scale, plot these numbers. On what kind of line do all the points lie? 13. The lengths of the sides and the areas in square inches of 12 squares are : Sides.. 1" 2" 3" 4" 5" G" 7" 8" 9" 10" 11" 12" Areas., 1 4 9 16 25 3G 49 64 81 100 121 144 Using any convenient horizontal and vertical scales plot these observations. Do the plotted points lie on a straight line? 14. Plot these data and draw a smooth free-hand curve through the plotted points : Date of Population or Date of Population op Cknsus tr.S.INHlLLIOSS Census U. S. IN MXIiLJONS 1790 3.9 1850 23.2 1800 5.3 1860 31.4 1810 7.2 1«70 38.6 1820 9.6 1880 50.2 1830 12.9 1890 62.6 1840 17.1 1900 76.3 Plot dates on horizontal, and populations on vertical, lines. Draw a smooth free-hand curve through the plotted points. By continuing the curve can you predict (tell) about what the population will be in 1910? Is the line through the plotted points a straight line? §80. Meaturmg By Hundredths, percentage 1. What is j^^ of 500 mi.? 4^ of 500 mi.? ^f^ of 500 miles? 2. What is 4^ of 200 A.? ^^ of 200 A.? ^Vd of 200 acres? Note.— First find jJo. 3. What is -^VV of 100 ft.? ■^%\ of 100 ft.? ^|j of 100 feet? 4. i of anything equals how many hundredths of it? 5. How many hundredths of anything are the following frac- tional parts of it? i'y iy i'i f; 1 5 l> f; i\j A> A; iV; «V5 ?V> zTfy /?• 124 RATIONAL GRAMMAR SCHOOL ARITHMETIC Definition. — ^One hundredth is often written 1% and read 1 per cent- The sign (% ) is a short way of writing "one one-hundredth." Per cent is a short way of speaking **one one-hundredth." 6. Bead and give the meaning of the following : 2%; 6%; 8%; 12%; 12^%; 25%; 33*%; 87*%; 100%. 7. Give the simplest fractional equivalents of (fractions that are equal to) these per cents : 8. How many lb. are 50%; of 8 lb.? of 18 lb.? of 24 pounds? 9. How many square feet are 25% of 16 sq. ft.? of 48 sq. ft.? of 88 sq. ft.? of 400 square feet? 10. Heating an iron rod 100 in. long increased its length 2%. How many inches was the length increased? 11. One boy threw a stone 100 ft., and another threw it 12% farther. How many feet farther did the second boy throw the stone? 12. Referring to Fig. 1^ (page 31) point out the following per cents of it: 20% ; 40% ; 60% ; 80% ; 2% ; 4% ; 10% ; 100%. 13. Draw a square and divide it by a line so as to show 50% of it; 25% of it; 75% of it; 12*% of it. 14. Similarly, show the same per cents using a circle; a rectangle. 15. Draw a line of any length and show 33*% of it; 66* % of it; 100% of it; 12*% of it; 10% of it; 20% of it. 16. Draw a square and divide it into small rec- tangles as shown in the figure. Point out the following per cents of it: FIGURE 64 25%; 33*%; 50%; 66*%; 75%; 12*%; 100%. 17. A pair of shoes costing $2 was sold at a gain of 50%. What was the amount gained? 18. A boy was 50 in. in height 2 years ago. Since then he has grown 4% higher. How many inches taller is he now? How tall is he now? 19. The height of a 14 year old boy is 60 in., and a girl of the same age is 2% taller. How tall is the girl? i^ y MEAStJREMEi^t 1S5 20. An umbrella was marked $1.50, and was sold at a reduc- tion of 20%. At what price did it sell? 21. Eighty per cent of the cost of a girl's suit is $4.00. What is 1% of the cost? What is the whole cost, or 100% of the cost? 22. An article sold for $4.00 after having been marked down 20%. What was the price before it had been marked down? 23. A lot was sold for $800, which was 60% more than it cost the man who sold it. Kow much did it cost him? Note.— The 60% means 60% of what it cost the seller. This cost is how many per cent of itself? $800 is, then, how many per cent of the original cost. 1% of the original cost equals what? 100% of this cost equals what? 24. A thermometer reading was 120°, which was 20% higher than the reading taken 10 min. before. What was the previous reading? Note.— In such questions as this answer first the question, "20% of what?" In this case 20% of the previous reading. 25. A man sold his farm for $12,000, which was 25% more than he paid for it. What did he pay for it? 26. This year a flat rented for $55 a month, which is 10% more than it rented for last year. For what did it rent last year? 27. Out of 80 games played by a championship team, 20 were lost. What per cent of the games were lost? What per cent were won? 28. The chest measure of a boy at the close of the school year was 27^ in., which was 10% greater than at the beginning of the year. What was the boy's chest measure at the beginning of the year? 29. A man paid $18 for the use of $300 for a year. What per cent was the sum he paid of the sum he borrowed? §81. Simple Interest. An extensive use of the system of measurement by hundredths is made with problems in Interest. Definition.— Jwferesf is money to be paid for the use of money. Two per cent interest means that 2^ is to be paid every year for the use of 100^ or $1 : $2 for the use of $100, and so on at this rate. What then would 6% interest mean? 4% interest? 10 fc interest? 126 RATIONAL GRAMMAR SCHOOL ARITHMETIC The amount of money to be paid each year for the whole sum bor- rowed is called the Interest for 1 year. 1. What is the interest on $200 at 4% for 1 year? 2. What is the interest at 6% on $1 for 1 year? Note. — In computing interest a year means 12 months of 30 days each. 3. What is the interest at 6% on $1 for ^ yr., or G mo.? for 2 mo.? for 1 month? 4. What is the interest at 6% on $1 for 3 mo.? for 5 mo.? for 7 mo.? for 14 mo.? for 16 mo.? for 34 mo.? for any number of months? 5. From the answer to problem 2, find the interest at 6% for 1 yr. on $8; on $25; on $48; on $85; on $124; on $450. 6. If you know the interest at 6% for 1 yr. on $1, how can you find the interest at 6% for 1 yr. on any number of dollars? 7. From the last answer to problem 5, find the interest at 6% on $450 for 2 yr. ; for 3 yr. ; for 4^ yr. ; for 12^ yr. ; for any number of years. 8. Using the third answer to problem 4, find the interest at 6 % for 7 mo. on $48; on $450; on $75. 9. Tell how to find the interest at 6% on any number of dol- lars for any number of months. 10. The interest on a certain sum of money for 1 yr. at 6% is $24. What is the interest on the same sum for the same time at 12%? at 18%? at 30%? at 60%? at 1%? at 3%? at 2%? 11. f of the interest on a sum of money for a given time atQ^o equals the interest on the same sum for the same time at what rate per cent? Note.— J of a number may be easily found by subtracting from it I of itself. How may i of a number be found similarly? 12. Knowing the interest on any sum of money at 6%, how can you quickly find the interest on the same sum for the same time at 7%? at 8%? at 11%? at 15%? at any rate per cent? 13. Find the interest on $1200 at 6% for 2^ yr. ; for 2i years. 14. A man has $350 in a bank, which pays 3% interest. To how much interest is the man entitled if his money has been in the bank 2 yr. and 4 mo.? 6 yr. and 9 mo.? 8 yr. and 2 months? COMMON USES OF NUMBERS §82. Pressure of Air. oral work 1. When a glass is filled level with water, covered with a piece of writing paper, and carefully inverted, why does not the water fall out when the glass is held month downward? 2. When a soft leather sucker is moistened and spread out on a smooth, flat surface, why does it cling to the surface even when the sucker is raised by being lifted at its middle? 3. When one end of a glass tube is placed in water and you draw with the mouth at the other, why does the water rise in the tube? 4. Fill a bottle with water, leave the cork out, and invert the bottle in a vessel of water. Why does the water not run out? 6. When a glass tube with one end closed is partly filled with mercury and the open end dipped under the surface of mercury in a cup, why does not all the mercury run out of the tube? Note. — In connection with 5, examine a mercury barometer. 6. When a sheet of thin rubber, or paper, is held over the mouth of a funnel, and the air is sucked out of the funnel through its neck, why does the rubber curve inward? Will it do this in all positions of the funnel? Why? These experiments show that the air presses downward, upward, and in all directions upon surfaces. Careful measure- ments have shown that the pressure of the air on every square inch of surface is about 15 pounds. Note. —This pressure is equal on all sides of surfaces, upper sides and lower sides, outside and inside, toward the right and toward the left. WRITTEN WORK 1. Measure the length and the width of the cover of your book and find the downward pressure in pounds on the upper surface of your book when it lies flat upon the desk. Why is the book so easily moved about under this pressure? 127 128 RATIONAL GRAMMAR SCHOOL ARITHMETIC 2. Measure and compute the downward pressure of the air on the top of your desk. Can you lift this number of pounds? Why can you lift the desk? 3. A room is 10' x 20' x 25'. Find the pressure of the air, in pounds, on the floor, on the ceiling, and on each of the four walls of the room. Why does not this pressure tear the walls apart? 4. The average surface of the human body equals 20.6 sq. ft. What is the total pressure of the air on the outside surface? Why does not this pressure crush the body? 5. Measure the length and the width of the door of your room, and find the air pressure on one side of the door. Why can you open and close the door so easily? 6. Measure and find the air pressure in pounds on the outside surface of a pane of window glass. Why does not this pressure break the glass? §83. Passenger and Freight Trains. 1. A fast train runs from Chicago to a station 356.4 mi. dis- tant in exactly 9 hr. What is the average rate (miles per hour) of the train? 2. The train left Chicago at 6: 10 a.m. At what time did it arrive at the station? 3. Another train runs 10^ hr. at an average rate of 36 mi. per hour, including stops; how far does it run? 4. If the train (problem 3) started at 2: 30 a.m., at what time did it reach the end of the run? 5. A traveler went by rail from Chicago to Los Angeles, California, in 4 da. 19 hr., at the average rate of 37.4 mi. an hour, including stops. How far is it from Chicago to Los Angeles by this route? 6. A train ran 361 mi. in 9^ hr. What was its average rate? 7. A freight train is running 21 mi. an hour while a brakeman on top of the cars walks toward the engine at the rate of 2^ mi. an hour. How fast does the brakeman actually move forward? how fast, if he walks from front to rear at the rate of 2^ mi. an hour? 8. A conductor walks from the front to the rear of a train at the rate of 3^ mi. an hour while the train is running 38 mi. an hour. COMMON USES OF KtJMBEllS 129 At what rate per hour does the conductor actually move forward? At what rate does he move in the direction of the running train if he walks from the rear to the front at the rate of 3^ mi. an hour? 9. How may the conductor (problem 8) suddenly change his rate of motion from 34^ mi. to 41^ mi. an hour? How many miles an hour does this change of rate amount to? 10. A passenger train running 32 mi. an hour meets a freight train running in the opposite direction on a parallel track 18^ mi. an hour. At what rate do the trains approach each other? 11. At what rate does the passenger train (problem 10) pass the freight if both are running in the same direction on parallel tracks? 12. A boy tries to overtake a street car by running up behind it. If the boy runs 10 ft. a second and the car runs 6 ft. a second, how soon will the boy overtake the car if he is now 20 ft. behind it? 13. Two friends, coming from opposite directions, have arranged to meet at a certain railway station. Their trains are running at 35 mi. and 25 mi. an hour. Not allowing for time lost in stops, how soon win their trains be together at the station, if they are now 30 mi. apart and both trains reach the station at the same time? 14. A passenger train is made up of a postal and baggage car. an express car, 3 common coaches, 2 chair cars, a dining car, and 2 sleepers. The average length of a car is 61|', and the length of the engine and tender together is 65'. How long is the train? 15. The weight of the engine (problem 14) is 142,780 lb.; of the tender, 43,200 lb.; and the average weight of a car is 83,480 lb. What is the total weight of the train, in tons? 16. To draw a train on straight, level track at a speed of 40 mi. an hour, requires a horizontal pull of t^^-^ of the weight of the train. What force in tons would be needed to draw the train of problem 14 at a speed of 40 mi. an hour? 17. An engine, 62' long and weighing 240,000 lb., draws a tender, 22' long, weighing 64,500 lb., and a train of 72 empty freight cars, averaging 36^' in length and 32,700 lb. in weight. How long is the train? how heavy? 130 RATIONAL GRAMMAR SCHOOL ARITHMETIC 18. If it requires y/^i^ of the weight of a train to draw it os straight, level track at a speed of 20 mi. an hour, how many poands of force must the engine of problem 17 exert to draw the train, on such track, 20 mi. an hour? How many tons of force? Note.— First find jAo of the weight of the train including engine and tender. 19. How much force would be needed to draw 38 cars, each weighing 16^ T., and each loaded with 18^ T. of coal, on straight, level track at a speed of 20 mi. an hour? (See problem 18.) 20. When a train is moving 5 mi. an hour it takes a horizontal force of about ii^j^ of the weight of the train to draw it along straight, level track. On such track what force must an engine exert to draw a train of 68 empty freight cars, each weighing 34,300 lb., at a speed of 5 mi. an hour? 21. Under the same conditions as in problem 20, what force would be needed to draw a train of 38 cars, each weighing 34,000 lb. and carrying a load of 58,600 lb. of coal in addition to its own weight? 22. A railroad company purchased 6 locomotive engines, weighing, in pounds, 142,780, 142,630, 158,670, 139,790, 146,890, and 138,960. The tenders weighed, in pounds, 43,750, 44,200, 45,280, 42,920, 43,650, and 44,280. The cost per pound of the combined weight was 13f <^. What was the total cost? 23. The greatest horizontal pulling force that an engine can exert on dry, unsanded steel track in starting a train is about f of the part of the engine that is carried by the driving wheels. What is the greatest horizontal pull the engine of problem 17 can exert in starting a train on such track, supposing the entire weight of the engine is carried by the driving wheels? 24. Answer a similar question for the engine of problem 15, supposing 105,750 lb. of the weight of the engine is carried by the driving wheels. §84. Train Despatcher's Report. A copy of a train despatcher's sheet, for a test run, showing the distance between stations, the schedule time and the exact mn- ning time of the train, the number of the cars in the train, the number of passengers carried each trip for 7 week days. COMMON USES OF NUMBERS 131 Pbbfobmancb op Tbaik No. 25, from July 1st to 8th, 1898. Sta- tion 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. DiB- TANCS Ml .. .. 3.1 .. 5.5 .. 7.9 .. 12.0 .. 17.0 .. 19.9 .. 24.5 .. 27.6 .. 33.8 .. 38.7 .. 43.5 .. 50.5 .. 53.8 .. 55.5 SOBSDXTLB TIMB 3:50 3:56 3:57 3:59 4:02 4:06 4:09 4:12 4:15 4:20 4:24 4:28 4:33 4:38 4:40 IBT 3:50 3:55 3:57 3:59 4:02 4:06 4:08 4:lli 4:13i 4:18 4:22 4:25i 4:301 4.32^ 4:35 J 2d 3:49 J 3:55^ 3:58 4:00 4:03} 4:07 J 4:10 4:14 4:16| 4:21 J 4:25 4:28} 4:34) 4:371 4:40 4th 8:50] 3:56 3:581 4:00} 4:03} 4:07} 4:09} 4:13 4:15 4:20 4:24 4:27} 4:32} 4:35 4:37} 5th 6th 7th 3:50 3:55 3:57i 3:59} 4:03} 4:06} 4:08} 4:12 4:14 4:18} 4:22 4:26 4:31 4:34 4:36 3:50} 3:55 3:585 4:00 J 4:04 4:08i 4:11 4:14} 4:17} 4:22 4:26} 4:30 4:35} 4:38 4:40} 3:50} 3:561 3:57} 4:00 4:03 4:07} 4:10 4:14 4:16 4:20} 4:24} 4:28 4:33 4:86 4:37} 8th 3:51} 3:56 3:58} 4:00 4:03 4:07} 4:10 4:13} 4:15 4:20} 4:23 4:27 4:32 4:35} 4:37 Number of cars Passengers carried Running time Mi. per hour (average) . . 5 201 7 441 6 11 5 79 5 107 113 6 118 PROBLEMS 1. By **running time" is meant the difference between the time of leaving* station 1 and the time of arriving at station 15. On a separate sheet write the * 'running time" for each day. 2. Find the difference between **schedule time" and the time the train actually reached station 6 on the 2d, 4th, 6th, 7th, 8th. 3. The train is **on time'' if it reaches a station just at "schedule time." Find how much th^ train was ahead of or behind time in reaching station 12 on each of the 7 da. Mark "ahead of time" results with an "A," thus: A. 2J min., and "behind time" results with a "B," thus: B. IJ minutes. 4. Make and answer similar questions for other stations. 5. How far is it from station 1 to station 15? from 2 to 9? 4 to 11? 3 to 15? Make and answer similar questions for other stations. 6. What is the difference in schedule times between stations 1 and 15? 2 and 9? 4 and 11? 3 and 15? any other two stations? • Stops are so short that leaving and arriving times are regarded as the same. in HATIOKAL OllAMMAR SCHOOL ARITHMETIC 7. From your answers to problems 5 and 6 find the average* rate of running (in miles per minute) between stations 1 and 15, by a train running exactly on schedule time ; between stations 2 and 9; 4 and 11 ; 3 and 15. 8. Find the average rates of running between the same stations (as in problem 7) on the 6th. 9. Find the average rates of running between stations 1 and 15 on each day, that is, the "mi. per hour." 10. What is the total number of cars drawn during the whole period? 11. Find the total number of passengers carried. 12. Make and solve other problems on the table. §85. Areas of Common Forms, oral work 1. If one side of an inch square represents 80 rd., how many square rods will the inch square represent? how many acres? 2. In a drawing of a rectangular farm to a scale 1 in. = 80 rd., how could you find the number of acres in the farm? 3. If a square field containing 40 A. is cut across diagonally by a- railroad, how many acres does each of the triangular parts contain, supposing the railroad itself covers 3 acres? wmw" ~ m i 1 mi'' \^' M mm 'm '■/,', ''/ly.'i.~hWX FIGUBB 65 4. Give the areas of the cross-lined portions of the 9 forms of Fig. G5. ♦ "Average rate" here means distance run divided by time of running. COMMON USES OF NUMBERS 133 PROBLEMS Each inoh in Fig. 66 represents 80 rods. 1. How many acres are in the tract ABODES What is the farm worth at $95 per acre? 2. The field is entirely enclosed, except along the river, by a barb- wire fence having 4 lines of wire. Barb wire weighs about 1 lb. per rod. Find the cost of the wire at $3.50 per bale of 100 pounds. 3. If posts cost 35^ apiece and are set 2 rd. apart, find the cost of the posts. FlOURB 07 4. In Fig. 07 1 in. represents 80 rd., or the scale is 1" to 80 rd. How many acres are in the field ABF? What is the field worth at $75 per acre? 5. Find how many acres there are in the field AFC and what is its value at $87^ per acre. 6. What part of the rectangle ADBF is the triangle ABF? What part of the rectangle EAFC is the triangle AFC? 7. What is the value of the field ABF at $80 per acre? 8. What part of the rectangular tract DBCE is the triangular field ABC? 9. How many rods of fence would be needed to enclose the field ABF? AFC? the field DBCE? 134 RATIONAL GRAMMAR SCHOOL ARITHMETIC 60rd. 10. A railroad runs straight across a rectangular field, dividing it into two triangles as shown in Fig. 68. How much is; each triangular part worth at $120 per acre, supposing the road takee of! equally from each part a strip containing 2^ acres? 11. The railroad company fences both sides of its strip with 4-wire fences, each 96 rd. long. Posts are set a rod apart, and cost 25^ each. The wire, used costs $3.25 per bale of 5ccl1b r= 40rd. 1^0 rd., and 2 men at $1.50 per day Figure 68 work 3 days to build the fence. What did it cost the company to fence its road? . 12. If 1 sq. in. of the rectangle of Fig. 69 represents f of an acre, how many acres are represented by the whole rectangle? 13. If one sq. ft. of a sheet of zinc weighs 2^ oz., how much will pieces of the same sheet of the sizes shown in Fig. 'J'O weigh? 14. If 15 sq.ft. ,12" Figure 69 te- - 16" >l of sheet copper weigh 2.5 lb., how much will 25 sq.ft. of the same sheet weigh? How many sq. ft. in 2.51b.? K — 36" — >♦ Figure 70 15. If 5 lb. of rope cost 75^, how much would 37^ lb. of rope cost at the same rate? 47 lb? 16. If the pendulum of a clock swings 40 times in 15 seconds, how many times will it swing in 1 minute? in 17 minutes? in 1 hour? 17. Two square feet of a 1-in. board weigh 9 lb. ; what would 9 sq. ft. of the same board weigh? 72 square inches? 16 square inches? INTRODUCTION TO FRACTIONS 135 INTEODUGTION TO EATIO AND PEOPOETION Ratio Definition. — The ratio of one number to another is the quotient of the first number divided by the second. 1. What is the ratio of A to 0? of to A? of B to A? of A to B? of 5 to 0? of to B? of Cto B? otCioA? of Cto 0? of to C? otDto C? otD to 5? of D to 0? of to 2>? .&'■■■ pr ■ &i FIGUBE 71 2. In the second square, compare each of the parts Ey Fy and with ; compare with each of these parts. 3. Compare each division with each of the others, separately. 4. In the third square, \ oi H \% how many times J? J of / is what part ot H? ^ of is how many times I? 5. What is the ratio of 3" to 1"? of 1" to 3"? of 1" to f7 of t" to 1"? 6. What is the ratio of 6 to 6? of 50 to 50? of J to j? of a to a? of a: to a;? 7. What is the ratio of any two equal numbers? ^ A ^ i is a short way of writing the following ' • t^^ expressions : FiouBB 73 The ratio of 1 to 4, 1-4, 1:4, 1 : 4 is read ''1 to 4," and i : 1 is read "i to 1.'' 8. What is the ratio of 1 ft. to 1 yd.? of 1 hr. to 1 da.? of 1 nickel to 1 dime? of 1 in. to 12 in.? of 1 ft. to 3 ft.? of 12 in. to 1 in.? of 1 mi. to 1 mile? s 13G RATIONAL ORAMMAH SCHOOL AHITHMETIO The ratio of a 12-in. line to a 1-in. line is called, also, the measure of the 12-in. line by the 1-in. line. 9. What is the ratio of 1 sq. ft. to a 3-in. square (see Fig. 73)? of 1 sq. ft. to 1 square inch? The ratio of a square foot to a 3-in. square is the measure of a sq. ft. in terms of the 3-in, square, that is, how many 3-in. squares in a square foot. Finding the ratio of the foot to the inch is the same as meas- uring the foot by the inch. Furthermore, to find the ratio of any number, or quantity, to any other number, or quantity, is to find the quotient of the first number, or quantity, divided by the second. The result of measuring one number by another is called the numerical measure of the first by the second. The ratio of one number to another, or the measure of one number by another, can be found only when both numbers are expressed in the same unit. 10. Measure the avoirdupois pound by the ounce; the foot by the inch; the inch by the foot; the ounce by the pound; the gal- lon by the quart ; the bushel by the quart. 11. Measure the rod by the yard; the mile by the rod; the square yard by the square foot ; the quart by the gallon. 12. Measure 8 ft. by 4 ft. ; IG ft. by 64 ft. ; a square mile by a square rod; an acre by a square rod; a square mile by an acre. 13. Measure 80 by 8; 80 by 4; 3 by 4; 4 by 3; 9 by 18; 18 by 9; 125 by 25. 14. Measure a by J; a by 2a; b by a; 2a by a; 6x by 2x; 2x by 6a:. 15. If 5 apples cost 4^, what will 35 apples cost? Solution. — How many fives of apples in 33 apples? How much does one live of apples cost? How much t)ien do 3o apples, or 7 fives of apples, costt This analysis may be thought of in the form of ratios. Cost of 35 apples x i_ . ^ 35 a5 .^. Fi — I — J— i — 1 — = T^f or more briefly -z = Tk 0) Cost of 5 apples 4f •'5 4^ ^ ' and we have to find a number which has such a ratio to 4^ as 85 apples has to 5 apples. This leads us to an equation of ratios. Definition.-— An equation of ratios is called sl proportion. INTRODUCTION TO FRACTIONS 137 §86. Proportion. In all these problems use the equation form of statement like (1) in the solution of problem 15 of the preceding section. 1. If 1 doz.. oranges cost $0.30, what will 4 oranges cost at the same rate? 2. If a yard of cloth costs $1.50, at the same rate what will ( yd. cost? 3. If a 3-in. square of tin costs i^, what will a square foot qpst at the same rate? 4. If 6 qt. of oil cost 15^, what will 3 gal. cost at the same rate? (4qt. = 1 gal.) 5. Two rectangular flower beds have the same shape. One is 3 ft. wide and 4 ft. long; the other is 6 ft. wide. How long is it? 6. Two books are of the same shape but of different siaes. One is 5" wide x 7^" long. The other is 15" long. How wide is it? 7. In two N N triangles of the a^ A "l? X ^/X^ same shape, like a/ \ b a \b " "* those of Fig. 74, / \ \ (1), i{a=4",^« ^ ^ ^ 12", and J = 4", "^ ^^^ how long is B? 8. In triangles of the same shape, like those of Fig. 74, (2), if a = 4", A =. 12", and * - 5", how long is B? If a = 6", b = 8", and B = 32", how long is A? If J = 7", B = 35", and A = 30", how long is a? 9. In triangles of the same shape, as those of Fig. 74, (3), if A = 21", b = 9", and B - 27", how long is a? If a = 5^", A = 22", and B - 30", how long is b? 10. In triangles of the same shape, like those in Fig. 74, (4), if a = 3", A = 9", and £7= 6", how long is c? It A ^ 24", c = 4", and C » 12", how long is a? 11. Denoting the sides of two triangles having the same shape by letters as in problem 10 ; if a = 8", b = 12", c = 17" and B ^ 30," how long are A and C? 138 RATIONAL GRAMMAR SCHOOL ARITHMETIC INTRODUCTION TO COMMON FRACTIONS §87. FractionB as Eatios and as Equal Parts. I. Into how many equal parts is A (Fig. 75) divided? What is one of the parts called? m m m m ^ F O H I J FlOVBB 75 3. One part of A equals how many of the smallest parts of B^ of C? of D? of E? otG? of /? of /? 4. One part of F equals how many of the smallest parts of G? of H? of I? of /? of Z>? of B? 5. Write the fraction that names one of the equal parts of -4, as divided in the illustration. 6. Measure A by one of its equal parts. Ans, 2 halves or f . 7. Measure the shaded part of B by one of its 4 equal parts. Ans, f, read "2 fourths." 8. Measure the unshaded part of B by the same unit. 9. Using one of the 8 equal parts of (7 as a unit, express the shaded part of in this unit; the right half of the shaded part. 10. Express E in terms of its smallest unit. II. Express each figure in terms of its smallest unit, as follows : (1) The whole square, F, G, H, /, /. (2) The shaded part of " " " '' " (3) The unshaded part of '* " " '* '' t^\ <( <c ^. <c cc . cc c< (4) The shaded part of G in the smallest part of 7; (5) The unshaded part of / and of G in thesmallest part of/. Definition.—} means 3 of the J's (one fourths) of some number, or •^ured quantity. The J is called the fractional unit. The fractional of any fraction is one of the equal parts expressed by the fraction. INTRODUCTION TO FRACTIONS 139 12. What is the fractional unit of each of the following frac- tions: f, I, f, 1, 4, h h h ii, h h h h h 1%. A. A, t'V? 13. What is the fractional unit of a fraction having the num- ber 7 below the line? the number 6? 9? 12? 15? 25? 18? 11? 64? 14. How many fractional units are expressed by the 1st frac- tion of question 12? by the 2d fraction? the 3d? 4th? 5th? 6th? 7th? Draw a square and divide it free-hand by lines showing the meaning of each of the first 8 fractions of problem 12. 15. What does the number written below the line of a fraction express? 16. What does the number written above the line of a fraction express? Definitions.— The number above the line is called the numercUor (meaning nitmbcrer). The number below the* line is called the denominator (meaning namer). 17. What are the fractional units of fractions having the fol- lowing denominators : - 3? 7? 12? 18? 24? 125? 75? 19? a? x? 18. Point out the fractions of question 12 that have the same fractional units. Definition. — ^The numerator and the denominator are together called the terms of a fraction. 19. If each half of a square is divided into 2 equal parts, into how many equal parts is the whole square divided? 20. Into how many equal parts is a whole divided, if each half of it is divided into 3 equal parts? into 4 equal parts? into 5? 6? 7? 8? 9? 10? 11? 25? 21. Into how many equal parts is a whole divided, if each third of it is divided into 2 equal parts? into 3 equal parts? into 4? 5? 6? 7? 8? 9? 10? 12? 15? 20? 30? 22. Into how many equal parts is a whole divided by dividing each of its fifths into 2 equal parts? 3 equal parts? 4? 5? 6? 7? 8? 9? 10? 20? 30? Give solutions of such as the following and explain: 23. Express as sixths, i, f , |, i^, |. 24. Express as twelfths, i, J, |, i, J, J, #. 140 RATIONAL GRAMMAR SCHOOL ARITHMETIC 25. In question 12 point out pairs of fractions that are not expressed in the same fractional unit, but may easily be so expressed. 26. Express the following pairs of fractions as equivalent fractions having the same fractional unit, or, what is the same thing, having a common denominator: J and f ; i and ^ ; | and f ; | and f ; f and |; i and -f^. 1 . 1 . 1 . 1 . 1 . t . 1 . 1 . 1 ^e 'A 'A ¥ 1 . . . 1 . . . 1 . , . i Vl2 ^ 'A 1 , 1 . 1 . 1 . 1 . 1 • 1 . t . 1 . 1 COMMON FEACTI0N8 REDUCTION OF COMMON FRACTIONS §88. To Eeduce Fractions to Higher, Lower, and Lowest Terms. ^^ 1. Express-Jas4:ths;8th8: 16ths. 2. Express J as 6ths ; 9ths 12ths; 18ths. 3. Express as 3ds f ; f y^ y^ 4. Express as 7ths /, ; H FIGURE 76 IJ. ||. 5. Compare the values of these fractions after reducing each to its lowest terms (that is, to the smallest possible whole num- bers for numerators and denominators) . 1 5 I J W> \\\ \\\ tVttj \%%' 6. Express! as lOths; 15ths; 20ths; 35ths; 75ths; lOOths. 7. By what must you multiply the numerator of f to make the numerator the same as that of |? By what must you multiply the denominator of \ to make the denominator the same as that of I? Draw a square and divide it free-band to show the com- parative sizes of I and | (see square (7, Fig. 75, §87). 8. By what must you multiply each term of | to obtain ^^ W f ^? tVi7? f H? What, then, is the value of each fraction of problem 5? 9. How then may the value of a fraction be expressed ia higher terms? 7 COMMOK FRACTIONS 141 A fraction may be expressed in- higher terms ^ without chang- ing its value^ by multiplying both terms by the same number, 10. Express the following fractions in higher terms: I; -l;l;il;il;T'j; If 11. How can you obtain the terms of | from the terms of f ? of ■^%? of IJ? of ^Vo? 12. What fraction do you obtain by dividing each term of j*g by 4. What, then, is the value of /^ as compared with the fraction ^? 13. By rising a rectangle, divided as in Fig. 77, show that ^^ of the rectangle equals \ of it. 14. In what lower terms can f ^ be expressed? fiqubb 77 T*A? iS^ Show that f J = f by means of a properly divided rectangle. 15. What effect on the value of a fraction is made by dividing each of its terms by the same number? 16. How may the value of a fraction be expressed in lower terms? Principle I. — Multiplying or dividing both terms of a frac- tion by the same number changes the form of the fraction without altering (or changing) its value. Fractions are most easily used when their numerators and their denominators are the smallest possible whole numbers. Definition. — The fractions are then said to be in their lowest terms. Among the many ways in which Jf can be written are these: 17. Show how } is obtained from J| by the use of Principle I. 18. Give the values of these fractions in their lowest terms: tV; I> Isj tV> II 'j ft- To obtain the value of a fraction in the smallest possible terms, it is necessary to divide both terms by the largest exact common divisor of both terms. Definition. — ^This divisor is called the greatest common divisor of the terms. It is indicated by the initial letters G. C. D. 142 RATIONAL GRAMMAR SCHOOL ARITHMETIC Dividing both terms of a fraction hy their 0, C, D. reduces the fraction to its lowest terms, 19. Why are these fractions in their lowest terms? t; i; *; i; I; !; \h if; i*. Among these fractions is there a factor common to any numer- ator and its denominator? , Definition. — Two numbers that have no common factor, except 1 , are said to be prime, to each other. 20. Reduce these fractions to their lowest terms : ii;ll; i-»; ll;.fl;T¥r; Hh §89. Factors, Prime and Composite. Note. — Review tests of divisibility §57, pp. 73, 74, and use them . through this section and the next, when searching for factors. 1. Write down all the factors, or exact divisors, of 36 and of 48. Definition. — By factor is here meant exact divisor, or a divisor that is contained without a remainder. Convenient Form 2 )36, 4 8 The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 2 )18, 24 18^ and 36. 3)9^ The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 2. What factors are common to both 36 and 48? 3. What is the greatest factor that is common to 36 and 48? 4. Arrange the factors of 42 and 105 as the factors of 36 and 48 are arranged in problem 1, and answer questions like 2 and 3 for the factors of 42. and 105. 5. Give the factors that are common to the 3 numbers, 18, 24, and 36. Give the greatest factor common to all three numbers. 6. Answer questions like those of problem 5 for these numbers : (1) 12, 72, and 84; (2) 42, 98, and 168; (3) 36, 84, 96, and 108. COMMON FRACTIONS 143 7. Are there factors of 5 other than itself and 1? of 7? of 13? Definitions. — A number that has no factors except itself and 1 is a prime number. A number that has factors beside, itself and 1 is a com,' posite number. 8. Name the prime numbers from 1 to 100; the composite numbers from 1 to 100. Query. — Is 2 a prime or a composite number? Definitions. — A number that can be exactly divided by 2 is an even number. All numbers that can not be exactly divided by 2 are odd numbers. 9. How can an even number be quickly recognized? (See test of divisibility by 2, p. 73.) 10. Name the even numbers from to 50; the odd numbers. 11. Tell what numbers of these are (1) even, (2) odd, (3) prime, (4) composite: 5, 8, 9, 2, 21, 15, 19, 26, 27, 38, 41, 42. 12. Mention some numbers that are both odd and composite ; even and composite ; odd and prime. 13. Factors of 28 are 7 and 4. As 7 is a prime number it is called a prime factor. What is a prime factor? 14. Write the prime factors of 21; of 24; of 25; of 27; of 30. Note. — Write out the prime factors of these numbers as they are here written out for 96: 96 = 2x2x2x2x2x3. 15. How many times does 2 occur as a prime factor in 96? This may be indicated by writing 96 thus: 2^x 3. The small 6 written to the right and above the 2 is to show how many times 2 is to be used as a factor. §90. Greatest Common Divisor by Prime Factors. 1. What are the prime factors of 36 and 48? (See problem 1, §89.) Write 36 in the form given for 96 in problem 15 of the last section. Write 48 also in this form. 2. Will each of the common prime factors of 36 and 48 divide the G. C. D. of 36 and 48? Will the product of all the common prime factors divide the G. C. D. exactly? 3. Answer questions like 1 and 2 for the numbers 42 and 105 ; for 96 and 21^ ; for 75 and 250. 144 RATIONAL GRAMMAR SCHOOL ARITHMETIC 4. When any common prime factor occurs oftener than once in one or both of the numbers, how often does it occur in the G. C. D. of those numbers? Answer by examining these pairs of numbers : (1) 12 and 48; (3) 75 and 250; (2) 54 and 405 ; (4) 98 and 343. 5. Make a rule for finding the G. C. D. of two or more numbers from their common prime factors when no common prime factors are repeated in any of the numbers. Test your rule by finding the G. C. D. of 30 and 42; of 105 and 231 ; of 30 and 70. 6. Make a rule that will give the G. C. D. of two numbers when one or more of the common prime factors is repeated in one or both of the numbers, and test your rule by finding the G. C. D. of 72 and 108; of 288 and 648; of 675 and 1125. 7. Find the G. C. D. of 792 and 1080. Convenient Form 2) 7d2 1080 792 = 2» X 32 X 11 1080 = 28 X 3a X 5 G. C. D. = 2^X3=^ = 8X9 = 73 Ti 3 )l5" 5 Principle II. — The G, C, D, of two or more mimbers equals the product of all the prime factors common to all the numbers^ each commo7i prime factor being used the smallest number of times it occurs in any one of the given numbers. 8. Find the G. C. D. of the sets of numbers in problem 6, §89. 9. Find the G. C. D. of the following, applying tests for divisi- bility: (1) 100 and 120; (5) 108, 162, and 216; (2) 54 and 144; (6) m, 176, and 286; • (3) 225 and 315; (7) 315, 630, and 756; (4) 215 and 1935; (8) 162, 1134, and 1458. 10. Reduce these fractions to their lowest terms by dividing both terms by their G. C. D. : 2) 396 540 2)198 270 3)99 135 3)33 45 (1) \%; (4)H«; (7) 1!?; (10) ^Vt; (2) if; 0) m\ (8)VA; (n).TRA; (3) n-, (6) m; (f) iSI; (12) M- COMMON FRACTIOKS 145 §91. froblems. In each case when the answer to the problem is a fraction, or ratio, it rriust be expressed in its lowest terms. 1. A man works 48 da. out of 64 da. What part of 64 da. does he work? 2. A grocer bought 24 boxes of oranges and sold IG. "What part of his purchase was sold? 3. Out of 56 bu. of potatoes a huckster sold 49 bu. What part of his potatoes was sold? 4. A street car on one line makes a weekly average of 72 trips. A car on another line makes an average of 144 trips. What is the ratio of the former to the latter? 5. A man pays $50 for wood and $125 for coal in one season. Find the ratio of the cost of the wood to the cost of the coal. 6. Out of 1000 ft. of lumber purchased, 500 ft. were used for the flooring of two rooms. Let x equal the ratio of the number of feet of lumber used to the total number of feet purchased. Find the value of x. 7. Out of 126 bu. of oats, a livery man fed 63 bu. in 1 wk. Let y equal the ratio of the quantity of oats bought to the quan- tity of oats fed. Find y. 8. In 667 lb. of sandy loam there were 377 lb. of sand and gravel. What part of the soil by weight was sand and gravel? 9. The human body needs about 94.5 oz. of water and solid food each day, of which 64.8 oz. should be water. The water is what part, by weight, of the totaV quantity of solid food and water? Suggestion. — Both terms of the fraction may be multiplied by 10 without chaDging the value of the fraction. Then find the G. C. D. of the new terms and divide both new terms by it. 10. The total population of a certain city is 36,720, of which 12,243 are colored and 10,017 are foreign. What part of the entire population is colored? What part is foreign? 11. What part of the total population is made up of colored and of foreign persons? The solution of problem 10 will show that J of the population is colored, and that -^j of it is foreign. AVe can solve problem 11 if we can find the value of y + -^i . 146 BATIONAL GRAMMAR SCHOOL ARITHMETIC 12. A boy spent f of his money and gave away J of it ; what part of it did he have left? To solve this problem we must know how to add and subtract fractions. This is what we shall study next. §92. Fractions Having a Common Fractional Unit (a Common Denominator). 1. Complete these equations : |yd. + |yd. = fgal. + fgal. = 6 + 6 " M^+tA = i^ + tV = ^ + ^ = X~ X |hr. + ihr. = A+A- Ob a+6 4 wk. + f wk. = jV + iV- z~ u 1 bu. + 1 bu. = 4r+4 = 2. + ^.= n ' n Note.— -J- is read, ' •5 divided bya,"and~ is read, "x divided by z. ^^ is read, *'a plus 6 divided by c." 2. Make a rule for adding fractions having a common denom- inator. 3. Complete these equations: *-!= 1-1= T'V-f'T= tA-*A = tV-^= 4-1= i-|= 43da._VSrda.- ab mn caB__ ax c '~' o ~' X X ~' r r " n n " Note.— ^ is read, "a minus h divided by c." 4. Make a rule for finding the difference between two fractions having a common denominator. 5. Denoting any two fractions having common denominators by 7 and —, state in symbols : Principle III. — The sum^ or the difference^ of any two frac- tions having common denominators equals the sum^ or the difference^ of their numerators^ divided hy the common denominator. ' §93. Fractions Easily Beduced to Common Fractional Vnit. 1. } equals how many 12ths? Sths? 16ths? ^Oths? 64thB? lOOths? COMMON FRACTIONS 147 2. jV hr. + i hr. = a; hr. What is the value of x? Solution.— /, hr. + ^ hr. = j} hr. = | hr. a? = f . 8. yV hr. - } hr. = y hr. What is y'^ 4. Find these sums and differences : Fractions, whose fractional units are not the same, must be expressed in a common unit before they can be added or sub- tracted. To do this we must find a fractional unit, whose denom- inator can be exactly divided by each of the given denominators. It will be the simplest always to select the fractional unit whose denominator is the least number that each of the given numbers will divide. Definition. — A denominator which is common to two or more frac- tions is called a common denominator. When this common denominator is the least number that can be found whicli may be used as a common denominator of the fractions, it is called the least common denominator^ and is written L. C. D. Before studying more diflBcult fractions we must learn how to find least common denominators. We begin by a study of multi- ples, common multiples, and least common multiples of numbers. 5. Five measures of last year's growth of twigs from the lower branches on the north side of an oak tree are: 5^", GJ", 4J", 5J" and 6J". What is the average growth? 6. From the lower branches on the east and west sides of the same tree the measures are: 6j", 4f", (3i", GJ" and Gf. AVhat is the average growth of twigs on the east and west sides of the tree? 7. For twigs on the south side of the same tree the measures are: 6 J", 6j", GJ", 8 J" and ^'\ What is the average growth of twigs on the south side of the tree? 8. From the middle branches of the oak tree 5 measures of branches from the north side are: 5i", 4i", 7^", 3j" and 5J"; of branches from the east and west sides are: 4J", 7^", 5^", 7 J" and 6f"; of branches from the north side: 5^", Gf, 7f', 6f" and 7i". How much greater is the average growth of twigs on the south side than the average growth of those on the north side of the tree? than twigs from the east and west sides? 148 NATIONAL OHAMMAR SCHOOL ARITHMETIC §94. Multiples. ORAL WORK 42 yd. is exactly measured by 7 yd., 3 yd., and 2 yards. $21 is exactly measured by $7 and $3. 56 pk. is exactly measured or divisible by 7 pk. and 8 pecks. 42 is a multiple of 7, 3, and 2. 21 is a multiple of 7 and 3. 56 is a multiple of 7 and 8. What are some of the multiples of 3 and 5; of 5 and 7; of 2 and 11? 1. What then is a multiple of a number? Definition. — A number that can be exactly divided by another number is called a multiple of the latter number, or a multiple of a num- ber is found by multiplying it by some whole number. Following is a list of multiples of 2, from 2 to 36, inclusive: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 3G. Following is a list of all the multiples of 3, to 36: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36. 2. Why is each number in the first row a multiple of 2? 3. Passing along the rows underscore the numbers that are the same in both rows. 4. In what way are these multiples of 2 and 3 different from the rest? Ans. They occur in both rows. 5. What is a common multiple of two numbers? Definition.— A number that can be exactly divided by two or more numbers, is called a common multiple of those numbers. 6. Rewriting the common multiples of 2 and 3 we have: 6, 12, 18, 24, 30, 36. Can yon supply a few more common mnltiples here without extending the rows above problem 2? 7. What number, besides 2 and 3, will exactly divide all the common multiples of 2 and 3? 8. On the blackboard write out rows of multiples of 3 and of 5, like those above, underscoring, or writing in colored chalk, the multiples that are common to both rows. 9. Will the least of the common multiples of 3 and 5 divide all the other common multiples? What is the least common multiple of two numbers? COMMON FRACTIONS 149 Definition. — The least common multiple of two or more numbers is the least number that is exactly divisible by each of the numbers. It is usually written L. C. M. for brevity- §95. Finding the L. C. M. WRITTEN WORK 1. Make lists of the multiples of 2, 5, and 7, and find their L. C. M. 2. In the" same manner, find the L. C. M. of 3, 4, and 5. 3. Note that all the given numbers in problems 1 or 2 are prime to each other. In such a case how can the L. C. M. be found from the numbers? TTie L. C, M. of two or more Clumbers, all prime to each other ^ is their product. 4. Illustrate the truth of this statement by two sets of num- bers of your own selection, one to contain two numbers, and the other three, the numbers of each set being prime to each other. 5. 4, 8, 12, 16, 20, 24 are multiples of 4; and 6, 12, 18, 24, 30 are multiples of 6. What is the least common multiple of 4 and 6? 6. This number is the product of 4 and 6 divided by what? 2 is the common factor of 4 and 6. 7. Are 4 and 10 prime to each other? What is their greatest common factor or divisor? 8. 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 are multiples of 4, and 10, 20, 30, 40 are multiples of 10. What are some common mul- tiples of 4 and 10? What is their least common multiple? 9. Divide the product of 4 and 10 by 2, their G. C. D. What is the result? Compare this result with the last answer to problem 8. The L, C. M, of two numbers not prime to each other is their product divided by their greatest common divisor {0. C. z).), 10. What is the greatest common factor of 15 and 20? What is the product of 15 and 20? How can you find the L. C. M.? 11. In the same manner find the L. C. M. of 8 and 10; of 12 and 24; of 15 and 25; of 24 and 60; of 40 and 36. 150 RATIONAL GRAMMAR SCHOOL ARITHMETIC §96. Shorter Process for Three or More Nnmbert. ORAL WORK 2. In a certain number there are 8 24'8; how many 12's are there in the number? how many 8's? how many 6's? 2. In a certain number, x, there are 15 6's. How many 3's are there in a;? how many 2's? 3. If 6 exactly divides a certain number, rr, what other num- bers also exactly divide xt 4. If any composite number exactly divides a number, y, what other numbers also divide y? Any multiple of a number is divisible by all the prime factors of that number. WRITTEN WORK 1. Find the L. C. M. of 160, 504, and 540. Solution.— 150 = 2x3x5x5 =2x3x5*. 504 = 2X2X2X3X3X7 = 23 X32X7. 540 = 2x2x3x3x3x5 = 2^ X3»X 5. The L. C. M. is a multiple of each of these numbers severally. By the above principle it must then contain the prime factors 2, 3, 5, and 7. But, since 25 = 5* is a factor of 150 the L. C. M. must contain 5 twice as a factor. Since 8, or 2', is a factor of 504, the L. C. M. must contain 2 as a factor three times. How many times must the L. C. M. contain 3 as a factor? How many times must the L. C. M. contain 7 as a factor? The number that contains just these factors and no others is 2x2x2x3x 3x3x5x5x7 = 2»x3»x5«x7 = 37,800. 37,800 is the least common multiple, because it contains the necessary factors and no others. Any other common multiple must contain all these factors and some other, and it would there- fore be larger than 37,800. What prime factors must be in the L. C. M. of three numbers? If a prime factor is repeated in one or more of the numbers, how often must it occur in the L. C. M.? Principle IV. — TJie L, 0, M. of two or more numbers is the product of all the different prime factors of all the numbers^ each factor occurring the greatest number of times it occurs in any one of the numbers. 3) 6, 18. 27, 36 2) 3, 6, 9, 12 3) 1, 3, 9. 6 COMMON FRACTI0K8 151 2. Find the L. 0. M. of these numbers and show your work : (1) 9,12,18; (4) 15,25,42,50; (2) 6,24,30; (5) 7,28,24,42; (3) 18, 30, 36; (6) 6,18,27, 36. 3. The work can be shortened by this arrangement. Let us solve problem 2, (6). ^ „ Explanation.— Select any prime Convenient Form number, as 8, that will divide two or more of the numbers whose L. C. M. is sought. Divide it into all the numbers that are exactly divisible by it, writing in the line below the 1, 1, 3, 2 quotients and any numbers the chosen T n ur q\/On/QvQ\xo -iaq divisor does not exactly divide. L. C. M. -3X2X3X3X3- 108. ^^^^ another prime number and do as before. Continue until the numbers last brought down are all prime. The continued product of the divisors and the numbers remain- ing in the last horizontal line is the L. C. M. A little study will show this to be merely a more convenient way of finding the factors described in Prinbiple IV. The least common denominator defined on page 147 is the L. G. M. of all the denominators of the fractions. 4. Find the least common denominator and then add (1) Ai A» A> and if; (3) i, -f^, /y, and ^^; (^) 4> ijVi A-> and ^ji • (4) |, V^^, j\, and ^V- §97. Definitions and Principles. Thus far we have had to do with two kinds of numbers: (1) whole numbers, or Integers^ and (2) fractional numbers, or j^ractions. Definitions. — A proper fraction is a fraction whose numerator is less than its denominator: as |, |. \. An improper fraction is a fraction whose numerator is equal to, or' greater than, its denominator : as |, |, I, f . A mixed number is a number, such as 2}, 12}, 7f , that is composed of an integer and a fraction. 1. Name the proper and the improper fractions, the mixed and the whole numbers : f, f, iV, 5, V-, I, I, 1^, 3xV, 6.75, ^^, 2. 2. Make and give an example of each class. 3. How many times is 1 contained in 3? in 18? in 25? in any number? 152 RATIONAL GRAMMAB SCHOOL ARITHMETIO We may write this in symbols, thus: 1 = 3; Y=i8; V=2S; 7-«. Any whole number may he written in the fractional form by writing 1 for its denominator. 4. Write the following numbers in fractional form: 12; 24; 32; 68; a; x 6. How many halves are there in 1? in 2? in 5? in 2J? in 5J? 6. How many 5ths are there in 1? in 2? in 4? in 6? in 6|? in 71? 7. Change the following integers into 6ths: 2; 5; 16; 29; 120. 8. Change the following numbers into 12ths: 2; 8; 12; 20; 56; 128. 9. How may any whole number be changed into 12ths? into 9ths? into 24ths? into a fraction having any given denominator? Principle V. — Any integer may he expressed in the form of a fraction having a given denomhiator hy multiplying the integer by the given denominator and writing the product over that denom^ inator. 10. Express 6, 18, 17, 22, 39, 28 as 4ths; as 7ths; as lOths; as lOOths. 11. Change into 6ths, 2; 4^; 24|; 37f 12. Change the following numbers into improper fractions having 7 for a denominator : 3;^; 24f; 106f; 234^; 648. 13. How can you change 'any mixed number into an improper fraction whose denominator is the denominator of the given frac- tion? Principle VI. — A mixed number may be expressed as an improper fraction by multiplying the whole number by the denom- inator^ adding the numerator to the product^ and writing the sum over the given denominator. 14. Reduce to improper fractions: H\ 6i; 12i; 18i; 25f ; 328|; 609^. COMMON FRACTIONS 153 t4^ 15. Express V- as a whole number. Change V- ^ * mixed nnmber. 16. Change to whole, or mixed, numbers the following improper fractions: f; J; J; V-; V-; V; ¥; V; V; If; It; V; V; i 17. How may any improper fraction be changed to a whole, or mixed, number? PRii;rciPLE VII. — An improper fraction may be changed to a whole^ or mixed^ number by making the indicated division. 18. Reduce to whole, or mixed, numbers the following: I; V-; If; n^; V-; IM; HF- §98. Addition of Fractioiis. 1. If the numbers written on the lines of the drawing are the lengths in feet of the inside walls of the house, how many feet of lumber would there be in a baseboard not over V thick, 1 ft. wide extend- "'- l^^ ing entirely around the inside of the house, deducting 6 board feet for doors? To solve this problem we add first all the '*^^ 12 j4 whole numbers and then all the fractions. We must now learn how to add such common fractions as occur here. This problem will be solved later. (See problem 9 below.) 2. A man owns f of an acre in one block and | of an acre in another block ; how much land does he own? This problem requires us to add f and i. I. Geometrical Solution.— Fig. 79 (a) is divided by the vertical lines into 3 equal parts and by the hori- zontal lines into 8 equal parts. Into how many e(][ual parts do the 2 sets of parallel lines divide the square (a)? Show by Fig. 79 (a) that i = 15- Show by Fig. 79 (5) that { = ii- r 9^ 2^4 FlOUBB 78 (a) (0) Fioniai ra | + f = H + H = H = Hi- Ans. \\\ aor99- 154 EATIONAL GRAMMAR SCHOOL ARITHMETIC n. ARiTtiMETiCAL SOLUTION.— What is the least common denomi- nator of f and f ? What, then, is the largest fractional unit (therefore having the least denominator) in which both } and i can be exactly- expressed? i = how many 24ths? I = how many 24ths? 164-21 _ ,,, . -,, Ana. m acres. Add 4, ij, f , and '^^. 7 12 *< (b) (c) (d) fi = A 1 = 1! A = H FlGUBB 80 (a) I. OE0METRICAL SOLUTION.— Show from Fig. 80 (a), that f = H ; from Fig. 80 (6), that A = A; from (c) that J = ?f ; from (d) that ft = fj. U + A -h S5 + fl = w = m = iii. II. Arithmetical Solution.— The least common denominator of the fractions is 84. 3X12, 2X4, 2X28, 5 X 7 _ 36 + 8 + 56 + 85 84 f + A+i+i% = 185 * 84 7X12^21 X4 ' 3X28 ' 12X7 = ili=lH. The last step consists in reducing the result to its simplest form. 4. ^ of the weight of a quantity of soil was gravel, and f was sand; what part of the soil hy weight was sand and gravel? Draw a figure and give the geometrical solution. Give also the arithmetical solution. 5. One side of a triangle is 2f in., another is 2f in., and the third is 3 1 in; how long is the perimeter of (distance around) the triangle? Add first the whole numbers, then the fractions, and finally add the sums. G. One side of a 4-sided figure is 6f in. long, a second side Is 3t in., the third Tf in., and the fourth T^V in. What is the perimeter of the figure? 7. A coal dealer bought 4 carloads of coal of the following weights: 3UJ T., 27| T., 29| T., and SOJ T. .Find the com- bined weight, in tons. COMMON FRACTIONS 155 8. Solve these problems as rapidly as you pan work accurately, asing your pencil merely to write the products and the sums : (1) I + * =? (4) 4+iH = ? (7)H + i = ? (2) i + i =? (5) H+tV=? (8) -^+1- = ? (3) A + A = ? (6) iV + if =? (9) |- + f=? 9. Solve problem 1 of this section. §99. Subtraction of Fractions, 1. A boy had $| and spent t^; what part of a dollar did he have left? Arithmetical Solution.— The least common denominator is 2 X 6 «10. l=A;l = iV Then, 1-1= ft- A = iV ^n«.$ft 2. William is 4f ft. tall and James is 4^^ ft. tall; who is the taller and by how much? 3. A man owned {i A. of land and sold f A. ; how much land did he then own? 4. From f lb. of loam f lb. sand was removed; how much of the loam remained? 5. A man having 14^ paid a debt of $2i; how much money had the man after paying the debt? Solution.— J = 5^, } = Jg. As ^ is less than If, write 4a^ in the form3|}. The problem is then 8iJ--2l8 = lA for 8 — 2=:1 and |J — 18 = A- 6. Solve these problems: (1) 13i - 7f = ? (4) 128f - 97if = ? (2) 284 -19f = ? (5) 639f -598jV = ? (3) 30^-16i = ? ' (6) 1217if -989i| = ? 7. A tree 72 J ft. high is broken off 28y\ ft. from the top; how high is the stump? 8. A 5-cent piece weighs 73} gr. and a quarter dollar weighs 96^ gr. ; how much more does a quarter weigh than a 5-cent piece? 9. The dime weighs 38j\ gr. Before 1853 it weighed 41J gr. By how much was the weight of the dime reduced in 1853? 156 rational grammar school arithmetic 10. A silver dollar weighs 412^ gr. and a double eagle weighs 516 gr. What is the difference between the weight of a double eagle and that of a silver dollar? 11. What is the difference between the weight of the half dollar (192^^ gr.) and that of the quarter dollar (96^1^ gr.)? 12. Write out rapidly the values of these sums and differences : (1) J+^ =? (6) 1 + f =? (2)i-iV=? (7)f-T\=? (3)+-T>j = ? (8)t\ + t\ = ? (*)^+V=? (9)-7+T=? (5)V-V=? (10)=--^=? 13. Make a rule for finding quickly the of any two fractions. Note.— This latter rule may be shown thus, 1st num. X 2d den. + 2d num (11) f + 1 = ? (12)t\-tV = ? (13) TV+l»tF-? (14)^+-? =? (15) ^-f -? sum, or the difference, X 1st den. and HiflpArAnruit — 1st den. X 2d den. 1st num. X 2d den. — 2d num. X 1st den. 1st den. X 2d den. The result must always be reduced to the simplest form. 14. In three successive runs, a train loses i, i, and i hr. How much time does it lose in all? 15. f of a man's salary is spent for clothing, i for board and lodging, ^ for books and stationery, yV for traveling expenses. What part of his salary was spent for these purposes? what part remained? 16. In the six days of one week, a man works 8^ hr., 9J hr., 8| hr., 9 J hr., 8f hr., lOf hr. What is the whole number of hours of work for the week? What are his wages at $.30 an hour? 17. Four children are to do a piece of work. Three of them do f , J, and f of it. What part is left for the fourth? 18. I, i, and -^j of a man's money are invested in three different enterprises. What part of his money is invested? What part is free? 19. A merchant sold y\ of a gross of buttons to one customer, and a of a gross less to another. What part of a gross did the second customer buy? COMMON PRACTIOKS 157 20. A owns yV and B ^^ of an estate. How much more of the estate does A own than B? 21. A tank of oil is f full. If ^ of the contents of the tank is drawn off and then -^ of the remainder, what part is left? 22. A owns ff of a mill. B's interest is J f of the mill less than A'b. What part of the whole does B own? 23. At $1 a day how much money does a man earn by working H da., ^ da., and 8| days? 24. A boy lives where school is taught 1100 hr. a year. He is compelled by sickness and other causes to lose from month to month the following numbers of hours : 115t; 39i; 15f ; 6H; 4f ; 3f ; 6|; 18J; 89^. How many hours did he lose in all? What part of the whole school year did he lose? Note. — In adding the fractions of problem 24 group together fractions whose denominators are the same, or are easily made t)ie same. §100. Multiplying a Fraction by a Whole Number. 1. 6 times 3 mi. = how many miles. 2. 8 X 4 yd. = ? yards. 8. 8 X 4 fifths = ? fifths. 4. 6x4 = —; what is the value of a;? 5. 9 X ^T = -—■; what is the value of a:? 6. A line is divided into 11 equal parts, and 5 of them are taken. What fraction represents the part of the whole which is taken? 7. What fraction would represent 3 times this part? 8. Replace the letter in each of these equations by its correct value; the sign (x) should be read "times" in. these problems. (l)6xf = f (5)5xHf (9)2xi\=^ (a)4xj=f (6)3x| = f (10)6x| = f (3)3xi = i (7)4xi = f (11)5x^1 = :^ (4)2xi = f (8)2x4 = f (12)2x^-:iV In multiplying these fractions by the whole numbers what change was made in the numerator to get the product? Make a rule for multiplying a fraction by a whole number. 158 BATIOKAL 6RAMMAB SCHOOL ARITHMETIC 9. Apply your rule to these problems, reducing to whole or mixed numbers all the products that are improper fractions: (1) 3xi= (5) llxA= (9) 6x 3 = (2) 5xf= (6) 9x,V= m 8xA = (3) 7 X f = (7) 4 x f = (11) 10 X I = (4) 9 X { = (8) 6 X } = (12) 9 X IJ = Note. — In such as (7) use cancellatiou, thus: i X «-= -^ = 3J. 3 10. Solve these problems and make a rule for multiplying a fraction quickly by its denominator: (1) 6 X i = ? (5) 12 X tV = ? (9) 75 X 4f = ? (2) 9xi = ? (6) 18x it =? (10) ^ X -f =? (3) 5 X t = ? (7) 21 X H = ? (11) wi X -^ = ? (4) 7xf = ? (8) 48x W=? (12) a: X -f =? Note.— In all such problems use cancellation, thus: /f X -s =5. 11. Give the values of the letter in each of these problems and make a rule for multiplying a fraction quickly by some factor of 7 7 its denominator; (use cancellation thus : ? X -jj = j- = If). 4: (1) 3x I =f; (3) 12x^V = f; (5) 25x^^ = f; (2) 5xA = f; (4) 9x/, = ^; (6) 13.x if =f State the rule. Principle VIII. — A fraction is multiplied by 1. Multiplying its numerator by the multiplier; or, 2, Dividing its denominator by the multiplier. Note. — The usefulness of the method of cancellation consists in male- ing these tvm processes t^ndo (balance) each other. (See problem II.) Query. — When will the second method (Principle VIII, 2), be the more convenient? A fraction is multiplied by its denominator by dropping Us denominator. COMMON FRACTIONS 159 §101. Mnltiplying a Mixed Number by a Whole Number. 1. What will be the cost of 12 yd. of muslin at 8f cents? Solution.— If 1 yd. costs 8f f what will 12 yd. cost? 12X8| means 12x8+13x1 = 96 + 8 = 104. Am, $1.04. Note. — In dealing with such expressions as 12x8 + 12x|. which contain both signs, {X)and (+), the indicated multiplications should be per- formed first, and then the additions. 2. 13 X 4| = ? 13 M 1Q N/ A Multiply the whole numbers in the usual way. 7i = 13 X I 13 X 8 = V = 7S, as in problem 8, §100. 59J = 13 X 4g PROBLEMS Solve the following, using the more convenient method in each case, employing cancellation whenever it shortens the work: 1. 5x|i= 5. 9xfJ= 9. 2xV- = 2. 3 X I = 6. 9 X I = 10. 3 X 6J = 3. 5x4|= 7. 9xyV= 11. 9x3^ = 4. 2x2i= 8. 9x f = 12. 8xy\ = 13. Find the cost of 25 yd. of cloth at *| per yard. 14. A earns $1|, and B earns 4 times as much. B earns $.r. Find X, 15. A grocer sold J bu. of apples to one customer, and 5 times that quantity to another. He sold y bu. to the second customer. Find y. If apples were $.40 a peck what amount of money did the grocer receive from the first customer? from the second? 16. If a man can cut J of a cord of wood in one day, how much can he cut in 6 da., working at the same rate? 17. One boy lives -^^ mi. from school and another 4 times aa far. What is the distance from school to the second boy's home? 18. Solve problems based upon the following facts: An iron tube weighs i lb. per foot. I of the weight of water is oxygen. The circumference of a bicycle wheel is 6J| feet. A west wind moved 47f mi. per hour. 160 RATIONAL GRAMMAR SCHOOL ARITHMETIC §102. Multiplying a Whole Number by a Fraction. The whole number is here the multiplicand. Definition. — To multiply a whole number by a fraction means to divide the multiplicand into as many equal parts as there are units in the denominator, and to take as man^ of these equal parts as there are units in the numerator of the multiplier. Illustration. — 12 multiplied by | means that 5 of the 6 equal parts of 12 are wanted. J times 12 = 2, and | times 12 = 5 X 2 = 10. g times 12 and | of 12 mean the same thing. 1. Solve these problems, reading the sign (x) "times": (1) f X 10 = ? (4) f X 15 = ? (7) t X 45 = ? (2) I X 25 = ? (5) f X 28 = ? (8) y«j x 26 = ? (3) i X 32 = ? (6) r\ X 33 = ? (9) t\ x ^7 = ? Suggestion.— In (9), ^f of 47 = 4^i, and 7 X4A = 28f} = 28+lH = 291{. Am. 29tJ. 2. Solve the following : (1) V^^ X 50 = ? (3) a X 105= ? (5) ^^ X 110 = ? (2) H X 39 = ? (4) 14 X 220 = ? (6) 6^ x 28 = ? Suggestion. — 6J x 28 means 6 X 28 + J X 28 = 168 + 21 J = 189{. Note.— See note to problem 1, page 159. §103. Factors May Be Interchanged. In Fig. 81. (a) the rectangle is supposed to be | in. wide and 5 in. high. The area is then 5 times | sq. in. = 2^ square inches. In (b) the rectan- a gle is 5 in. long and ^ in. high. Its area is i times 5 sq. in., or — ' " i of 5 sq. in. = 2^ sq. in., as is plain from Fig. 81. This shows the truth that when a fraction and a whole number are to be mul- tiplied it makes no difference in the nu- FiGUBi 81 merical value of tho product which of the factors we regard as the multiplicand. The name of the result is learned from the nature of the problem. ^j OOMMOK FRACTIOKS 101 Letting f- (read *'a; divided by y") stand for any fraction, and a stand for any whole number, state the following principle in symbols : The product of a tvhole number by a fraction^ or of a fraction hy a whole number ^ equals the product of the whole number by the numerator^ divided by the denominator, PROBLEMS 1. Find the cost of 2 J yd. of cloth at $3. 2. A man had $75; he spent f of it for a bicycle, and ^ of the remainder for clothing. Find the amount spent, and the amount he had left. 3. A gallon contains 231 cu. in. How many cubic inches in f gal.? in I gallon? 4. A cubic foot of granite weighs 170 lb. How many pounds in 5f cubic feet? 5. y\ of a plot of ground containing 918 sq. rd. was fenced for a garden. The garden contained how many square rods? 6. Make and solve problems, based upon facts obtained by yourself, or upon those given below. An avoirdupois pound of gold is worth about $348f . A bale of cotton ordinarily weighs 450 lb. Price 17^^ per pound. A tank contains 126 gal. linseed oil. Price $.62 per gallon. Corn is quoted and sold @ 50|^ a bu. Wheat is quoted and sold @ 77t^ a bushel. Oats are quoted and sold @ 32f ^ a bushel. To plow an acre with a plow cutting a furrow 10" wide (a 10" plow) a horse must walk 9y\ mi. ; with a 12"-plow, a horse must walk Si mi. ; with a 15"-plow, 6f miles. For a team drawing a plow, a distance of 16 mi. is a fair day's work and a distance of 18 mi. is a large day's work. The follow- ing table shows to what fractional part of an acre a furrow ^ mi. long and of the stated widths is equivalent: Width OP Furrow Equivalent 12" A A. 13" iA A. 14" <fy A. 15" ft A. Width op Furrow Equivalent 16" irVA. 17" -iVa A. 18" A A. 20" i«A 1G2 RATIONAL GRAMMAH SCHOOL ARITHMETIC (i) U04. Multiplying a Fraction by a Fraction. 1. What is the area of a rectangle 3"x 5", Fig. 82 (1)? 6"x 8"? T'x 8'? 2 rd. X 80 rd.? 16x20? a x J? 2. How many square feet in the area of a rectangle f x 8'? f 'x 12'? f x 16'? VV'x 30'? (Use cancellation. ) 3. What is the area of a rectangle f'xf'? [See Fig. 82 (2).] Solution.— -.4 J5Ci) is a rectangle J" long and I" wide. AEOH is a square inch. What part of a square inch is ABKHI What part of ABKH is ABCD ? Point out on the figure the part that represents } of | of 1 square inch? Into how many small rectangles, such as a, do the dotted lines divide the square inch? What part of the square inch is one of the small rectangles, a? How many of these are there in the given rectangle ABCD ? The area of ABCD is then what part of a square inch? We may write the two expressions for the area equal, thus: i X 4 = A. We see then that, if our law for finding the area of the rectangle, which we have found to hold for integers, is to hold for fractions also, we must define both i X 4 and f X | to be the same as | of }. Each is equal to ^s' It is clear that, whatever the fractions to be multiplied may be, the square unit [sq. in. in Fig. 82 (2)] may be divided by the cross lines into as many equal parts as there are units in the prod- uct of the denominators, and that there will be as many of these equal parts in the given rectangle as there are units in the prod- uct of the numerators. Hence our fractions may be multiplied by merely writing the product of the numerators over the product of the denominators with the dividing line between. Cancellation should be used whenever possible. 4. Draw a square inch and find these products as above: (1) t" X f " = ? (4) I" X f " = ? (7) V' X r ^ ? (2) f X i" = ? (5) r X f " = ? (8) f X f " = ? (3) r X f " = ? (6) r X f " = ? (9) f " X i" = ? COMMON FRACTIONS 163 5. By drawing and dividing a square inch, as in Fig, 81, show that the following equations are true : (l)4xA = 4ofA; (3) fx,«, = Aof4; (5)4xA = H (2),«ix 4 =Aof 4; (4) Ax? = f of A; (C) A><4 = H To find \ of 7^, we take a fractional unit which is \ of -f:^ (the fractional unit of 7^), and use the same number (5) of them, thus obtaining VV- B«t ^ of VV = 6 x 4 of VV = ^ x ^\ = 4^. As 4 X i^r = f ^f V^T> 4 ^ iS = f T' Similar reasoning would show also that Vt ^ T = f ^- Now we recall that the numerator (30) of the product (rfxf) was obtained by multiplying the numerators (6 and 5) of the factors {^ and 4). How was the denominator (77) of the product obtained? 6. State a rule for quickly multiplying a fraction by a fraction. 7. Solve* these problems as rapidly as you can work accurately: (l)'fxi (5)fxi (9)ixf (13) |xf (2)ixi (6)ixf (lO)txt (14)fx| (3)fx| (7) f xf (ll)fx| (15) |xf (4) |xf (8) 4xJ (12) 4x1 (10) |xf 8. Letting -| and y denote any two fractions, state the fol- lowing principle in symbols : 77ie product of any two fractions is a fraction whose numer- ator is the product of the numerators and ivhose denominator is the product of the denominators. §105. Multiplying a Mixed Number by a Mixed Number. 1. How many square feet in the area of a rectangle 4J" x 5^"? (1) How long is a (Fig. 83 ) ? how wide? What is its area? (2) Answer similar questions for J, (?, and <Z.. (3) How long is the entire rectangle? how wide? What is its area? figure 83 Since the area of the entire rectang:le equals the sum of the areas of its parts we may write (see note, problem 1, page 159): 4jX5i = 4x5+}X5 + 4X J + JX i = 20+ 3| + 2 + J = 261. A71S. 26t sq. ft. * Cancel whenever it is possible. ^4 b d 4 a c 5 Vz 104 RATIONAL GRAMMAR SCHOOL ARITHMETIC Point out on figure the areas that represent all the parts of the product. 2. Find the cost of 9^ T. hard coal at $7J. First Solution.— If 1 T. costs §7|, 9jt T. wiU cost 9^ x $7f. Convenient Form Explanation 7} 91 X 7} means 9 X 7} + 1 X 71 i X 7| means i of 7}. 63 =9 X7 6} = 9 X J = Y 3|= 1X7 1= 4X 5 9 X7f = 69} iX7J= 3J 9iX7J = 73| 73| Ana. |73|. Second Solution.— 91 T. = V T. ; 17? = IV Y X V = «!•. or 731. Ans. $73|. Find areas of surfaces having the following dimensions; 3. 44^ ft. X 28f feet. 4. 36| ft. X 27f feet. 5. 24f ft. X 18| feet. 6. 45f ft. X 30J feet. 7. 27i ft. X 18f feet. Find the cost of the following items: 8. 24f doz. eggs @ $.16|. 9. 46Jgal. of oil @ $.12i. 10. 15i yd. of cloth @ $.66f . 11. 52i Ih. of sugar @ |.05|. 12. 265f M. of pine flooring @ $35^. Make original problems from the following items: A cubic inch of water weighs 252f f grains. A water tower is 2|J ft. higher than the mound upon which it is built. The mound is 81| ft. high. G9^ statute miles = 1 degree of longitude at the equator. A pine tree 87 ft. high has no branches for ^f of its height. 1^ of a library containing 55,447 volumes was destroyed by fire. COMMON FRACTIONS 165 Flaxseed cost $1| per pound when a certain linseed oil factory bought supplies. A spring furnishes 28^\ bbl. of water daily. A certain vessel sails 11 J mi. per hour, on an average. A room is 32 ft. long and 24| ft. wide. Painting costs $^ per square foot. A cubic foot of water weighs 62.5 pounds. Gold is 19i times as heavy as water. 30^ sq. yd. = 1 square rod. §106. Dividing a Fraction by a Whole Number. ORAL WORK 1. What is i of 40A.? 40A. divided by 8 equals what? 2. \ of 35 lb. = ? 35 lb. + 7 = ? 3. tV of 80 = ? 80 + 10 = ? 4. i of 3 fourths = ? 3 fourths + 3 = ? 5. i of f A. = ? f A. ^ 5 = ? 6. J of 4 ft. = ? 4 ft. -4- 3 = ? 7. iot fjin. = ? fjin. -*-9= ? 8. Compare these fractional units, or unit fractions: (1) i is what part of i? (7) ^V is what part of i? (2) I is what part of i? (8) j\ is what part of J? (3) I is what part of i? (9) J is what part of I? (4) tV 18 what part of i? (10) yV is what part of i? (5) I is what part of i? (11) xV is what part of i? (6) i is what part of i? (12) -g^ is what part of i? 9. How is the size of a unit fraction changed by multiplying its denominator by 2? by 3? by 4? by 10? by 100? by a? by m? 10. By what must you multiply the first fraction in each of these pairs to get the second? (1) I and I? (4) V-i and « ? (7) ^\\ and |?? (2) ^ and i? (5) || and jj? (8) rk and f? (3) A and f? (6) |J and ff? (9) j^ and /,? 11. Make a rale for dividing a fraction by 3 without changing ^tt numerator; by 5; by 28; by a. 1G6 RATIONAL GBAMMAE SCHOOL ARITHMETIC WRITTEN WORK 1. What nnmber should stand in place of x in these examples? (1) i?-. 3 = /^; (4) V/-- 16 = t\; (7) ff*9 = fT; (2) -V * 15 = ? ; (5) -VV - 25 = j%; (8) ¥ H.a= I- ; (3)W-14 = fi; (6)41^- 144 = fT; (9)^*c=^. a. Solve these problems by finding what number should stand in place of x : (1) 1*3 = 4; (3) H- 8 = il; (5) 4H* 3 = l|i; (2) 1*4 = ^; (4) V*25 = -f^; (6) |H-15=ii!-. 3. In what two ways may a fraction be divided by a whole number? When would you use the method of problem 1? of problem 2? Principle IX. — A fraction may he divided hy a tvliole num* heVj (1) by dividing its numerator hy the wliole numher^ without changing its denominator^ or (2) hy multiplying its denominator by the whole number^ without changing its numerator. Note.— The note after Principle VIII, page 158, applies here alaa PROBLEMS 1. Divide f lb. of shot equally among 5 boys, and give the weight of one share. 2. i lb. of maple sugar is to be equally divided among 3 chil- dren. What part of a pound will each receive? 3. f of a piece of work can be done in 8 da. What part of it can be done in 1 day? 4. Make and solve problems based upon the following items, showing work in each case : %\\ to buy lace costing $5 a yard. $iJ to buy apples costing $3 per barrel. \ of an estate divided among 4 heirs. 5. If f lb. of coffee is divided into 4 equal portions, how much is there for each portion? 6. 8 lb. of butter cost If; lib. cost how much? COMMON FRACTIONS 16? 7. A strip of tin-foil | ft. long is to be cut across into 4 equal pieces. What is the length of one strip? 8. 4 children have a garden containing f A. If the children receive equal portions, what part of an acre does each receive? 9. 4 men own f of an estate, equally. What part belongs to each man? If the estate is valued at $3G,000, how much money is represented by each part? 10. If J^P A. is divided equally among 7 men, how many acres are there for each man? §107. Dividing a Mixed Number by a Whole Number. 1. 7 strips of carpeting cover a room S^^yd. wide. What is the width of one strip? 5i yd. = V yard. y yd. -*- 7 = i yd., the width of one strip. 6i is what kind of number? Before dividing what change was made? How was the division performed? 2. A boy earned $4^ in 6 da. What did his earnings average per day? 2 How was the dividing done in this case? 3. Make a rule for dividing a mixed by a whole number. To divide a mixed mimber by a whole number first change the mixed number to an improper fraction and then proceed as in the division of a fraction by a whole mimber, PROBLEMS !• 86f yd. of cloth were made up into 6 ladies' suits. If the suits contain the same number of yards, how many yards are there in each suit? 168 RATIONAL ORAMMAR SCHOOL ARlTHMfiTIC 2. A ceiling 16 ft. wide contains 233^ sq. ft. Find the length. 233i ft. =^^ feet. 175 " ^"^ ■*■ ^^ = 3^1^" '^^^ " ^^^^' -^^'"^ " ^^' '^" ^^^^^ ^® ^*^® length 4 of the room. 3. Solve the following problems : (1) 14i bu. -^ 8 = (6) $12f + 5 = (2) 9^Vyd.- 6= (7) 45^.5 = (3) 161 hr. -^ 12 = (8) ^^ + 4 = (4) 44yV Ib.^ 9 = (9) 14^V ^ 7 = (5) 39f mi. ^ 7 = (10) 66f -^■ 8 = Note. — In problems like (7) and (9), in which the integral part of the dividend exactly contains the divisor, divide the whole number and the fraction separately and add the quotients. 4. Find the lengths of the surfaces whose areas and widths are given here : Areas Widths Lengths (1) 212^ sq. ft. 17 ft. (2) 2Gli " 14 '* (3) 406i " 25 '* (4) 164| " 13 " Make problems based on these items: 5. A ship sails 246y^^ mi. in 36 days. 6. A man paid $195i for 72 days' board. 7. A field contains 2172J sq. rd. One side is 40J rd. long. 8. 15 lb. of fresh salmon cost $2.73|. §108. Dividing Any Number by a Fraction. (A) Dividend a whole number. 1. How many fib. packages can be filled from 3 lb. tea? (1) How many i lb. packages can be made from 1 lb.? How many J lb. packages can be made from 1 pound? (2) How many f lb. packages can then be made from 3 lb. tea? COMMON FRACTIONS 169 Solution. — J is contained in 1, 4 times. } is contained in 1, i this number of times. } is then contained in 1, } times. How many times is it contained in 3? Or, since f X | = 1, f is contained in 1, | times. 2. A boy earns at the average rate of $t per day selling papers. How many d3.ys did it take him to earn $12? (1) What effect does it have on a fraction to multiply its numerator? to multiply its denominator? (2) What effect does it have on a fraction to multiply both its numerator and its denominator by the same number? (B) Dividend a fraction. 3. Divide f hy 4. Solution.— 4x7 4x 7 |=47|'^*^y^ [Seeprob.2(2)above.] i^ = ^;why? 7X5 7X5 (C) Dividend and divisor both mixed numbers. 4. I paid $49| for coal @ $7| per ton. How many tons did I buy? Solution.— 248 31 -i^ 49J=^; 7J=^; 49|H-7f=-^ 8 ^8 -|-X?X4 _A L - 3 2 _ « 2 ^ ^X 5 X ^ Ans. 6| T. Let US now seek a general method of dividing any number by a fraction. 5. How many times is each of these fractions contained in 1? 170 RATIOKTAL GRAMMAR SCHOOL ARITHMETIC 6. How many times is each of these fractions contained in 1? t; I; 4; i; I; tV; tV; if; ii> H> tW> -s- 7. What number should stand in place of the letter x in each of these products? (1) fx| = a;; (4) Jx f =x; (7)T»jxV=a;; (2) fxi = aj; (5)T»,xV- = a;; (8) -| x ± =a;; (3) fx| = x; (6)|fxH = a;; (9) ^ x 4 =a:. 8. What number should stand in place of the letter y in these products? (1) |xy = l; (3) ,Vxy = l; (5) ffxy = l; (2) fxy = l; (4) Jfxy = l; (6) ■^xy = l; 1 divided by | equals f , and 1 + |=|; l->-8 = i and 1 + ^ = 8 and so with other numbers. Definition. — Dividing 1 by any number, whole or fractional, is called inverting the number. The reciprocal of any number is the number inverted. 9. To what number is the product of any number and its reciprocal equal? 10. Examine your answers to problem 7 and state how to find quickly the number of times any fraction is contained in 1. 11. If a fraction is contained in one | times, how many times is it contained in 2? in 6? in 12? in i? in }? in i? in any number? in a? Note.— 7 may be written f. 12 may be written Vi ^J^d so on. Con- sequently 7 and f are reciprocals, as are also 12 and ^^ ; 25 and A, and soon. 12. What is actually done by inverting the divisor in a problem in division of fractions? (See Definition above.) 13. If, then, the reciprocal of a fraction is multiplied by 6, what operation is performed? 14. Can you make a rule for quickly dividing any whole num- ber by a fraction? Principle X. — The reciprocal of any fraction shows how many times the fraction is contained ml. ' Principle XI. — Any number ^ mixed^ integral^ or fractional^ may be divided by a fraction by multiplying the number by the reciprocal of the divisor. I I.I COMMON FRACTIONS 171 15. Why do we invert the divisor? Why do we mnltiply the reciprocal of the divisor by the dividend? PROBLEMS 1. f yd. of silk costs $f . Find the cost of 1 yard. 2. At $1J a yd. how much cloth can be bought for $f ? 3. At $6|V per hundred how much beef can be bought for «2i? 4. How many meals will 4^ bu. of oats supply, if y\ bii. are fed at a meal? 5. How many pieces of wire ^ * \ i ft. long can be cut from a piece ' ' ' ' -^ 49ift. long? l^-r 6. Measure If ft. by f ft. PiGUBB 84 - (See Fig. 84.) Draw lines and illustrate the following : 7. 2i ft. + f ft. « 10. 3f ft. -H f f t. = 8. ^ ft. + A 't- = 11- li ft. -*- i ft. = 9. 2| ft. + 1 ft. = 12. Divide 3i yd. by i yard. 13. Divide 16* yd. by 2i yd. ; by If yd. ; by 4^ yd. ; by 1^ yd. ; by f yard. 14. Divide $6f by $tV; by $A; ^1 H\ by $J. 15. Find the number of strips of carpeting running the long way of the room : Width of Rooms Cabpetino 22i ft. ^ ft. wide. 7fyd. I yd. - 18i ft. ^ ft. ** 8i yd. I yd. *• 15ift. . 2i ft. *' Make and solve problems based upon the following items: 16. Portland cement $2} per bbl. ; $9 purchasing fund. 17. Sidewalk having an area of 607| sq. ft.; flagstones 5^ ft. square. 18. 15§ gal. maple syrup; fgal. measures. 19. 29f doz. oranges; ^ doz. oranges. 172 RATIONAL GRAMMAR SCHOOL ARITHMETIC 20. The ratio of two numbers is f; one of the numbers is f. 21. A man bought a house for $1860. The house cost him f as much as he paid for a mill. What was the cost of the mill? 22. My library contains 1075 volumes, which are f as many as a friend's contains. How many volumes are in my friend's library? 23. Of a number of cars of grain inspected on a certain day 86 were rejected. This number was J| of the entire number inspected. Find the number. 24. Let T- and — denote two fractions, and state the foUow- h y ing principle in symbols: The quotient of one fraction by another is equal to the product of the dividend by the inverted divisor. (Compare Principle XI.) §109. Complex Fractions. Any problem in division of fractions may be written in frac- tional form, the dividend being written above and the divisor below the line. Example. — The problem, to divide f by ^x ^^^J be written: Or, the problem, to divide 18| by 6|, may be written: 6| • Definition. — Fractions that contain fractions in one or both terms, are called complex fractions. Principle XI, §108, will enable us to solve all such problems. If either the divisor or the dividend, or both, are mixed numbers, they should first be reduced to improper fractions. Following this principle : ^-♦^¥ = H- The second step becomes unnecessary if we notice that the product of the outside terms (6 and 11), called the extremes^ gives the numerator, and the product of the inside terms (7 and 5), called the meansy gives the denominator of the result. COMMON FRACTIOKS 173 A complex fraction is simplified when its terms are freed of fractions, and the resulting fraction is expressed in its simplest form. In many cases factors may be cancelled from both numerator and denominator before multiplying the fractions. Cancel when- ever possible. Show that the method explained is the same as dividing the product of the extremes by the prod act of the meaiis in the given complex fractions. PROBLEMS Simplify the following complex fractions, cancelling when pos- sible : 1. 3. 4- 1 1 4. 6. * 4 6| L 3 15| T 10. 11. 12. 13. 84f 62| 14 M 15. §110. Joint Effects of Forces. ^/' B T^^r^T^ ^??r^?u "^^^ j^ ISI Figure 85. 1. The rear car, C, pulls back on the middle car, P, with a force of 100 lb., and the front car, A^ pulls it forward with a force of 250 lb. If car, ^, were taken out of the train whal; single for- ward force would draw it the same as it is being drawn in the train? 2. A force of 23 lb. is pulling the car (Fig. 9C ) toward the right and another of 12 lb. is pulling it "^ ^iQi^^rjQL . J *' toward the left. If the car is free to move, in FiGURB9a which direction will it move? What single force would move the car in the same manner as do both of these forces pulling at the same time? 174 RATIONAL GRAMMAR SCHOOL ARITHMETIC ' 8xib. 3. What single force will produce the tilWnmTOfi't^ s^me motion of the car (Fig. 97) as 82; lb. FiGUBBQ? acting toward the right and 5x lb. acting at the same time toward the left? ^^=^' r I I -rno ^^^ 4. What single force will move the car FIQURB08 (^^S- ^^) '^ ^^® same manner as 6m lb. pulling against 27n lb. will move it? When a force of 18 lb. is supposed to be pulling a car toward the rights it will be written thus : R 18 lb. When the same force pulls toward the lefty it will be written thus : L 18 lb. When two forces, R 16 lb. and L 12 lb., act at the same time, their joint effect is the same as the effect of the single force, R 4 lb. 5. What would be the joint effect of R 28i lb. and L21i lb.? 6. What would be the joint effect of each of these pairs of forces : (1) R 48i lb. and L 12f lb.? (4) R 23^ lb. and L 68 ^V lb.? (2) R 75yV lb. and L 69 ,V lb.? (0) R 16^58 lb. and L 81^ lb.? (3) R 181i lb. and L 129^^ lb.? (6) R 87f lb. and L 6841 lb.? A force of 25 T. (tons) , -pulling forward^ may be written F 25 T. > and a force of 18 T., pulling backward^ may be written B 18 T. The joint effect of both would be written thus : F 7 T. 7. What is the joint effect of the three forces: F 17 T., F 12 T. and B 25 T.? 8. What is the joint effect of each of the following sets of forces : (1) F 80^ T., B 28A T., and B 96f T.? (2) F 49i T., B 84| T., and F 248 jV T.? (3) B 68y\ T., F 73| T., and B 12^ T.? (4) F 28^% T., F 18i T., and B 37^^ T.? (5) F 692i T., B 386f T., and B 307^ T.? (6) F 2Sx T., F dx T., and B 36a; T.? 9. A force of 182 lb. pulling upward may be written U 182 lb. How would you write a force of 78 lb. pulling downward? 10. What is the joint effect of each of the following sets of forces : (1) U 8f lb., D 16^ lb., and U 3f lb.? (2) U 16i lb., U28 lb., and D 29^ lb.? COMMON FRACTIONS 1T5 (3) D 29 oz., D 32i oz., and D 16f oz.? (4) D 29f oz., U 65f oz., and D 42^ oz.? 11. If we call a force of 16 lb. pnlling forward, or upward, or eastward, or northward, or to the right, a positive force, and write it + 16 lb., we should call the same force pulling backward, or downward, or westward, or southward, or to the left, respe^- ively, a negative force. How should we write it? 12. Give the joint effect, with proper sign (+ or — ) of each of these sets of forces : (1) 2 forces, each -f 8 J lb., and one force, — 12 J lb. (2) 3 forces, each 4- 12J oz., and 4 forces, each — 8 J oz. (3) 5 forces, each - 16f T., and 12 forces, each + 9 J T. (4) 18 forces of - 16^^ lb. each, and 20 forces of + 25^ lb. each. §111. Exercises for Practice. 1. Find the following sums and differences as rapidly as you can work correctly. (1)4 + Vt = ? (7)8U4^=? (2)i|-i = ? (8)94-3|J = ? (3) H-A = ? (9) 16i-10^ = ? (4) if + il=? • (10) 210VV + 49VV=? (5)!|-il=? (11) 37i-18/T = ? (6) H + IJ = ? (12) 10U-98tV = ? 2. Find the products and quotients as rapidly as you can work correctly, using cancellation when it makes the work easier. (1) ixH = ? (9) 18f x9f = ? (2)Ax|f = ? (10)18^*161=? (3)HxH = ? (11) 21f*U^ = ? (4)U-/t = ? (12)6*15=? (5)il-A = ? (13) 8i| * 124 - ? (6) ^f--,»ir = ? (U) 154*3i = ? (7) l|xA = ? (15) U*6| = ? (8) 8f + 3| = ? (16) 261i + 80S = ? 3. Find the simplest results for these exercises: (l)ix|xT«r-? (3) (Vnx8i)H.6i-? (8)*Xt«^xH-? (4) (94x4f)*3i-? (5) ,»,+ tV + H = ? (6) * + *-« = ? (7) A X TVxif = ? (8) 1*1 + tV + H = ? (9) tS- f + f = ? (10) m -3J + 6i=? 176 BATIOKAL GRAMMAR SCHOOL ARITHMETIC (11) I6f-h70f-87i = ? (12) 16| X 3t X If = ? (13) (9f^6i)x6| = ? Note— First divide 9f by 61 and then multiply the quotient by 6§. (14) (224 ■*• Hi) X 19U = ? (15) (18i + 27f)+24t = ? §112. Dividing Lines, and Angles. Problem I. — Divide the line AB into 2 equal parts, see Problem VI, p. 101. EXERCISES 1. Draw a straight line with a ruler on the blackboard and Disect it with crayon, string and ruler. 2. Bisect each of the 3 sides of a triangle and connect each mid-point with the opposite corner of the triangle. These. lines are the medians of the triangle. How do they cross each other? 3. How might a line be divided into 4 equal parts by repeating this method? Divide a line into 4 equal parts by this method. Problem II. — Divide the angle BAD into 2 equal parts. Explanation. — With any convenient radius and with the pin foot on A {vertex) draw arcs 1 and 2 across AB and AD. Place the pin foot on the crossing point at 2, and with a radius longer than half way from 2 to 1 draw an arc 3. Now place the pin foot on 1 and with the radius used for arc 3 draw arc 4 across arc 3. » , «i .J, Call the point of crossing C Angle Bisected ^j^j^ ^j^^ ^^^^ ^^^^ ^^^ bisector AC. PiGUBB 09 Then angle BAC = angle CAD, EXERCISES 1. Draw an angle on paper, or on the blackboard, and with pencil, or crayon and string, bisect the angle. 2. Draw a triangle and bisect each of its 3 angles. How do the bisectors of the angles cross each other? 3. The line CD of Fig. 35, p. 101, drawn from C to D, is called the perpendicular bisector of the line AB. Draw a triangle of 3 unequal sides, and then draw the 3 perpencMcular bisectors of its sides. How do these lines cross each other? COMMOK FKACTIOKS §113. The Parallel Ruler. — A very good parallel ruler may be made by catting out two strips of cardboard or of very thin wood, exactly alike, as shown at (a), Fig.lOO. Thenumbeni show the widths and the lengths. To use the ruler place the two parts with their Fong sides together, the thin ends being in opposite directions as in the cut. Another sort, which the pupil may make for himself or pur- chase for a few cents, is shown at {h) , Fig. 100. The strips 8 may be of light cardboard or of thin wood of the lengths and the widths shown in the cut. The outside edges of these strips should be made as smooth and as straight as possible. The cross strips, which may be narrower than the strips Sy should be of exactly the same length between the pins at Cy Z>, and at E^ P. The distances EC and PD between the pins, should also be made equal. Common pins may be stuck through and bent over ^ W _ to hold the strips to- **-• gether at (7, 2), E^ and ^ _ ' ~ P. ^' " "'*"^ Z The pupil should provide himself with a parallel ruler for the following problems : FiGUBB 100 (h> FiGUBB 101 Problem I. — Draw a line parallel to a given line. Explanation. — Let AB be the given line. Fig. 101 (a) shows how this is done with the first kind of ruler, by holding the part 2 on the line, and sliding part 1 along, drawing the lines along the uppei edge of part 1. Fig. 101 (6) shows how to solve the problem with the second kind ol ruler. 178 RATIONAL GRAMMAR SCHOOL ARITHMETIC Problem II. — Draw a line parallel to a given line and through a given point. Explanation. — Sup- X. 2) P?s© ^^ (Fig. 102) is the given line, and P the point. Hold one edge of the parallel ruler along AB^ raise the other strip until its edge goes through the point P, and draw a line CD along this edge. PiGUM 102 CD is the desired line Problem III. — Solve Problem II with ruler and compass. Explanation. — First Step: Place the pin foot on the given point 1* and spread the feet until the pencil foot reaches some point, as C, on the line AB, Fig. 103. Draw the arc C2. Second Step: Place the pin foot on C and using the same radius as before, draw arc PU cutting AB9XD. Third Step: Spread the feet of the comnass apart as far as from P to D, ana placing the pin foot on C, draw the short arc a. Connect the crossing point E with P. EP Fiouaa 108 is the desired parallel to AB through P. EXERCISES 1. Draw a line on paper, mark a point not in the line and with a parallel ruler draw a line through the point and parallel to the first line. 2. Solve Exercise 1 on the blackboard with chalk, string, and ruler. 3. Draw a triangle on the blackboard. Draw a line through each corner of this triangle and parallel to the opposite side. Problem IV, — Divide a line AB into 3 equal parts (or trisect AB). Explanation. — Draw an indefinite line ACy making amr convenient angle with AB, Measure off 3 equal spaces from A to- ward O. Connect D with B. Draw through the points 2 and 1 lines parallel to T>B (see Problem II). Then AE^ Line Trisected ^^ = ^^' *^^ ^ ^"^^ ^ *''® **^® triseotioD Figure IM points. COMMON FRACT10K8 179 EXERCISES 1. Draw a line on the blackboard and trisect it. Note.— Draw the parallels by the method of Exercise 3 under Prob- lem III above. 2. Prom the suggestion of Fig. 105 draw a line _^.^C on the blackboard and divide it into 5 equal a*=*=^^-^^^-^B parts. FiGTJM 105 3. How may a line be divided into 7, or 9, or 13 equal parts? Draw a line on the blackboard and divide it into 7 equal parts. 4. How does the line connecting the opposite corners of a square divide the square? 5. Answer a question like 4 for the rectangle ; for the parallelogram. 6. If, then, the area of a square equals the FiouBs 106 product of its base and its altitude, what is the area of one of the 2 equal triangles into which a diagonal divides the square? Rect-dngle III bflag i?7 ^r^Ti^ Figure 107 7. Answer similar questions for the rectangle; for the parallel- ogram. 8. The area (A) of a rectangle of base h ft. and altitude a ft. is how many squarQ feet? 9. What is the area of a right-angled triangle (a right triangle) of base h in. and altitude a inches? 10. The base of a parallelogram is h in. and the altitude is a in. ; what is the area? 11. The base of a triangle is h and the altitude is a; what is the area? 12. The bases of a rectangle, of a parallelogram and of a tri- angle are b in. and their altitudes are a in. Find the ratio of the area of the rectangle to the area of the parallelogram ; the ratio of the area of the parallelogram to the area of the triangle. 180 RATIONAL ORAMMAB SCHOOL ARITHMETIC ^114. Vm8 of the 30'' and the 46'' Triangles. Pboblsm V. — To make the triangles for use in drawing. Explanation. — (a) Fold a piece of smooth heavv paper, having one straight edge (like the piece shown in Fig. 108), over a line near the middle. Bring the straight edges carefully together as shown in Fig. 109. FIOUBB 106 FlGUBB lOQ Crease the paper smoothly with a ruler or a paper knife and paste or glue the two pieces together. When the paper is dry, mark off distances of 4" from the square comer on the crease and on the straight side. Connect the 4 in. marks and cut the paper smoothly along the connecting line. This will give a triangle of the form T, Fig. 110. (b) In the same way, fold, crease, and paste another piece of paper a little larger than before. On the straight side mark off a distance CA equal to 3". With compasses, or with a stringer ruler, mark a point, B, on the crease so that AB equals 6''. Draw AB and cut out the tri- angle fif. Fig. 111. 4" FlOUBB 110 FlGUBB 111 If preferred the triangles T and S may be made of thin wood. Exercise.— Place the side, AC, of S against the long side of T, and holding the triangles witli the left hand, draw a line along AB. Now hold T, slide S a little (say \") and draw another line along AB. Simi* larly, draw a third line. Lines in such positions are called paraiW. lines. OOMIION PRACTI0N8 181 Problem VI. — Through a given point with a triangle draw a line parallel to a given line. Explanation.— AB is the given line and P is the point the parallel is to pass through. Placje a ruler CD, Fig 112, in such a position that when the triangle S is placed against it, one of the sides of 8 will lie along AB. Press the ruler against the paper and hold it with the left hand; with the right, slide the triangle alon^ the ruler until its side just touches the point P. Draw line FE through P and along the edge of the triangle. FE is ^the desired parsdlel line. FiGUBB 112 FIGUBB 113 Fig. 113 shows how this problem is solved with the triangles alone. EXERCISES 1. Solve the first exercise of Problem III with the triangles 8 and T as shown in Fig. 113. 2. Draw a triangle with sides of 1", IJ", and 2", and through each corner draw a line parallel to the opposite side. Use the ruler and the triangles. To draw the triangle see Problem IX, p. 103. 182 RATIONAL GRAMMAR SCHOOL ARITHMETIC Problem VII. — At a given point on a line with the triangles draw a perpendicular to the line. Explanation.— Let AB be the line and let P be the given point. FlGUBB 114 FIGURB 115 Holid one of the triangles, as T, in the position shown in Fig. 114, and^ placing one side of the other triangle, 5, against the upper side of T, slip 8 along until the square corner conies to the point P. Hold S firmly and draw the line PD along the side of S. PD is the required perpendicular. Problem VIII. — Through a given point not in a line, with the triangles draw a perpendicular to the line. Explanation.— Let AB be the line and let P (Fig. 115) be the given point. Hold one of the triangles, as T, so that its side lies along AB and slide the other triangle, S, with its side against the upper side of T until it oomes up to P. Then draw PD. PD IS the required perpendicular. EXERCISES 1. Draw a line, mark two points on it 1" apart and draw a per- pendicular to the line at each of the two points (by Problem VII) . Mark a point on one of the perpendiculars 1" above the given line and at this point draw a third perpendicular completing a 1" square. 2. Draw a triangle and through each corner draw a perpendic- ular to the opposite side (by Problem VIII). How do these per- pendiculars cross? 3. Draw any circle, also a diameter, marking its ends A and B. Through its center draw a perpendicular radius and prolong it, making a diameter, CD. Connect AC, CB, BD and DA^ forming an inscribed square. COMMON FRACTIONS 183 Scaler: 6 Scate r:20' §115. Scale Drawings of Familiar Objects. 1. Notice the scale of the drawing of the playhouse (Fig. 116), and compute the lengths of the following dimensions : (1) The width; (2) the height of the lower side; of the higher side; (3) the rise (dif- ference of higher and lower sides); (4) the height of the door; the width ; (5) the distance from the right side of the door to the right corner of the house. 2. From the dimensions in Fig. 116, with ruler and triangles, make a drawing of the playhouse to a scale 4 times as large as that of the drawing (Fig. 116). 3. From the scale of the drawing (Fig. 117) find the following dimensions of the house : (1) Tbe width; (2) the height of the eaves ; (3) the rise {ah) ; (4) the width and the height .of the door; (5) the width and the height of the window. 4. Find the area of -the end of the house, including the gable and excluding the areas of the door and the window. 5. Make an enlarged drawing of the house to a scale B times as large as the scale of the drawing (Fig. 117). 6. From the scale of Fig. 118, give the length and width of the door; the length and the width of the panel. (Find the length by measurement). 7. Similarly, give the length and the width of the door B, also the length and the width of the upper panels, and the widths of the strips enclosing the panels. 8. Make an enlarged drawing of each of the doors to a scale 5 times as large as the scale of the drawings (Fig. 118). FiGUBB IIV 184 RATIONAL GRAMMAR SCHOOL ARITHMETIC 9. HofV^ long are the cross-pieces and the strings of the kite represented by the drawing (Fig. 119)? 10. Make an enlargec? drawing of the tite to a scale 8 times as large as that of Fig. 119. 11. Notice that the horizontal piece divides the surface of the kite into two trapezoids. The alti- tude of the upper trapezoid in the drawing (Fig. 119) is li'\ *^^ ^^*^ of ^^^ lower trapezoid is fj". How many square feet of paper are needed to cover the kite? (Allow 144 sq. in. for folding and pasting over strings.) 12. What is the length of each leg of the chair shown in the drawing (Fig. 120)? What is the height of the back? the depth of the seat to the extreme rear? 13. Make a drawing of the chair to a scale 5 times as largo as that of Fig. 120. 14. Find from the drawing of the gate (Fig. 121), (1) the height of the high end of the gate; Sc*h n40^ ^ (2) of the lower end ; (3) the FiGUBB 120 length of the long brace; (4) the length of the gate; (5) the width of the strips. 15. Make a drawing of the gate to a scale 8 times as large as that of Fig. 121. iU»"- V6' 16. Fig. 122 is a scale drawing of the figurb 121 side and the end views of a large book. How long is the book? how wide? hovr thick? 17. Make an enlarged drawing of the book to a scale 4 times as large as that of Fig. 122. 18. From your own measurements make a drawing, to any convenient scale, of a thick object, as a block, a brick, a crayon-box, showing two views as in Fig. 122. 19. Make a scale drawing from your own measures of a desk, table, bookcase, or other object in your schoolroom, showing three different views (top, side, and edge views) of it. KHo View* Figure 123 COMMON FRACTIONS 185 §116. SchooUiouse and Oronnds. 1. Using a foot rule, graduated to 16ths of an inch, and regarding the scale of the drawing (Fig. 123), find the width of the grounds; the length; the area in square rods. (30^ sq. yd.= 1 square rod.) 2. Find the length of the field; the width; the area in square rods. 3. Find the length and the width of the school yard ; the area in square rods. 4. Find the length and the width of the schoolhouse ; the area, in square yards, cov- ered by it. 5. How far is the front door of the schoolhouse from the front fence? from the west front gate? from the sand pile? from the tree? 6. How far is it from the back door to the east flower bed? to the back fence? to the west fence? to the coal shed? to the north- east corner of the school yard? to the south end of the pond? to the hiU? to the nearest point on the creek bank? to the foot bridge (F)? 7. How wide is the south road? the creek? the branch? 8. How many square rods in the south road in front of the grounds? in the crossing of the roads? 9. How many square rods in the meadow? in the grove? in the pasture? 10. How many square rods are covered by the creek and the branch together, within the fence lines? ROAD 1 r Settle n 160' Figure 123 186 RATIONAL GRAMMAR SCHOOL ARITHMETIC 11. How many rods of fence will be needed to encloBS the grounds and to ran along the lines indicated? 12. From your own measurements make a similar drawing, to a convenient scale, of some tract of ground (school yard or field) near your schoolhouse. Locate any fixed objects on your tract in their proper places on the drawing by measuring their shortest distance from a fence and from a comer. §117. Proportion. oral wore 1. Compare the ratio 3 : 2 with the ratio 6 : 4. What do you find? 2. Compare the ratio 8d. : 2d. with the ratio 16 men: 4 men. 3. Compare the ratio 10 ft. : 800 ft. with the ratio 100 mi. : 8000 miles. 4. A proportion is an equation of ratios. Thus 6 : 12 = 3 : 6 and A = f *^re two different ways of writing the proportion. Definitions.— The first, second, third, and fourth numbers of the pro- portion are called the first , secondly third, and fourth terms of the propor- tion. The first and fourth terms are called the extremes, and the second and third terms are the means. The first two terms are the first couplet, the third and fourth terms are the second couplet, 5. In 6 : 12 = 3 : 6, to what is the product of the extremes equal? the product of the means? 6. Answer the same questions for 3 : 5 «= 12 : 20. 7. Which of these pairs of ratios may form proportions: 4 : 9 and 8 : 18? 6 : 11 and 18 : 33? 1:3 and 8 : 21? 3 : 7 and 12 : 28? 6 : 11 and 12 : 22? 1 and j\^? SL and ,^? « and S.? ^ and Jg-? la 2b ax e» n 6n 8. Is 4 = i\ a proportion? Multiply both sides by 21 and we have 4 x 21 = 15. Now multiply both sides of this equation by 7 and we have 5 X 21 = 7 X 15. What were the terms 5 and 21 in the proportion called? What were the 7 and 15 in the proportion caiiti? RATIO AND PKOPORTION 187 9. Compare the product of the Ist and the 4th numbers in these proportions with the product of the 2d and the 3d terms: (1) 4 = il ; (5) t\ = M ; (2)A=i?; (6) i-U.'^ (3)H= fl; (7) -^«H-;; Wl =V/; (8) f = *-*f Can you state the principle problems 8 and 9 illustrate? Peinciple. — In a proportion the product of the means equals the product of the extremes. WRITTEN WORK 1. What must x be in each of these expressions to give a pro- portion? Solution of first equation: 2x = 15, or a; = 7.5. 1. 2:3»5:a; 2. 4:3 = 8:a; 3. 6: a; = 9: 27 4. 7:2 = a;:14 5. a;: 6 = 8: 12 6. a:b = 2a:x 7. 18: a; = 54: 18 8. 4:6= a;: 9 10. ?* = i bo e 2. What does the letter stand for in each of these proportions? (1) - = - ^''' 7 85 (4)1 = 1? ^ ' 11 TO W 7-30 11-1 a 12 o^ 16 (6) ^'^ 10 "45 («)il °36 (Q^ a: _ 36 . V"'' 18 "324' 5 13. '66' (11) , 11- ^. 44" 8' — - (12) ^'^^f 161-14' (16) m' 15 _£. 18 ~ 4' 188 RATIONAL GRAMMAR SCHOOL ARITHMETIC §118. Practical Applications. 1. At the rate of 20 lb. for $1.00, what will 15 lb. of sugar cost? 2. A boy buys 36 oranges at the rate of 3 for 5 cents. What do they all cost him? Note. —Write the problem in the form f = — and find c, the cost of the'oranges. Use Principle § 1 1 7. 3. The boy sells his 36 oranges at the rate of 2 for 5 cents. How much does he receive for them? 4. A man walks 54 miles at the rate of 9 mi. in 2 hr. How long does it take him to walk the 54 miles? 5. A number of boys buy 72 marbles at the rate of 6 marbles for 5 cents. How much do the 72 marbles cost? 6. If it takes 5 yd. of ribbon to trim 4 hats, how many hats can be trimmed with 37^ yards? 7. At the rate of $4.00 for 5 dozen, what will 3 doz. cocoanuts cost? 8. In a class there are 3 girls to every 4 boys. If there are 24 boys in the class, how many girls are tl^ere? 9. At the rate of $7.00 for 3 days, how much money does a man earn in 21 days? in 27 days? in 16^ days? 10. If it takes 7 yards of gingham to make 4 aprons, how many aprons will 42 yd. gingham make? 56 yd.? 594- yd.? 11. If f of a ton of coal is worth $4.81, what are 12 T. worth at the same rate? Note. — Call the unknown term x. Write the proportion, using a?, and then use the^Principle, §117, to find x. 12. A spelling class of 52 pupils writes a total of 1352 words; at the same rate how large a class will write a total of 728 words? 13. A girl jumping a rope makes 441 skips in 3 min. How long will it take her to make 392 skips at the same rate? 14. If the average column in a newspaper contains 1600 words, how much should a writer receive for 700 words at the rate of $5 per column? 15. In going 10,725 ft. the front wheel of a bicycle revolves 1430 times. How far would it go in making 1001 revolutions? RATIO AND PROPORTION 189 _6£ECh eftOOK, 16. A clock ticks 7 times in 5 sec. ; how many times does it tick in 2 da. 6 hours? 17. If 60 A. cost $3000, how many acres will cost $2450? 18. The line BC (Fig. 124) represents 20 rods. FE is twice as long, AF four times as long, CD and ED iBve times as long, and AB six times as long. Find the length of each side. 19. Find the value of x in the following proportions: EC :FE=AB:(x); BC : (x) = FE : CD; AF:FE= (x) : BC, 20. When a foot rule is held Figure 134 2^ ft. (arm's length) in front of the face as shown in Figure 125, 7 in. on the ruler seems just to cover the edge of a door 7 ft. high. If the ruler is held parallel to the edge of the door, how far is the eye from the door? 21. If the shadow cast by a 4-ft. stake is 5 ft. long, how high is a tree which casts a shadow 50 ft. long on the same day and hour? 22. The shadow cast by a 3^ ft. stake is 10| ft. long. How high is a flagpole that casts a shadow 225 ft. long at the same time? 23. How high is a house that casts a shadow 120 ft. long at the moment when a stake 6 ft. high casts a shadow 20 ft. long? 24. A road runs 2 miles south, then 1 mile west, then f mi. south) then 1^ mi. southeast, and then 2^ mi. south. In a scale drawing of the road the two-mile part is represented by a line 18 in. long. How long a line will represent each of the other parts of the road in the same drawing? 25. What is the scale of the drawing just mentioned? 26. What is the scale of a drawing in which a distance of 20 ft. '%ft"^ -— "1 ^ f Figure 125 [ 190 RATIONAL GRAMMAR SCHOOL ARITHMETIC is represented by a line 2| in. long? By a line | in. long? By a line 2i ft. long? ' 27. The food parts of beef are water, fat, and protein. In 1 lb. of loin steak there are 3| oz. of water, 2| oz. fat, 2| oz. pro- tein, and 2^ oz. of waste material. How much of each of these parts are there in 3 lb. of loin steak? In 5J lb.? In a 12-pound loin steak? 28. In 4 lb. sirloin steak there are 34| oz. water, lOf oz. fat, lOf oz. protein, and 8y^y oz. waste matter. How much of each is there in an 8 -lb. sirloin steak? In 24 lb. sirloin steak? In 30 pounds? 29 In a 10-lb. porterhouse steak there are 5^ lb. water, If lb. fat, If lb. protein, and IJ lb. waste. How much of each is there in a 30-lb. porterhouse steak? In 15 lb.? In 18 lb.? In 48 pounds? 30. In 15 lb. of beef ribs there are GJ lb. water, 3} lb. fat, 2J J lb. protein, 3} lb. waste. How much of each is there in 10 lb. beef ribs? ' In 25 lb.? In 10 lb.? In 75 pounds? DECIMAL FRACTIONS §119. Notation of Decimals, oral work 1. In $1111 what does the 1st 1 on the right stand for? the 2d? the 3d? the 4th? 2. In $6666 what does the 1st 6 on the left denote? the 2d? the 3d? the 4th? 3. In $372.68 what is the unit of the 3 (Aiis, $100)? of the 7? of the 2? the 6? the 8? 4. In $5555 how does the number denoted by each 5 compare with the number denoted by the 5 to its left? to its right? 5. How do the units of the places of a number change as we pass through the number from left to right? 6. Which unit is the fundamental unit out of which the other units are made? How is the place of this fundamental unit indicated in $675.28? Definition. — A dot, called the decimal point, or point, is used to show the units' digit. The point always stands just to the right of the units' digit or place. DECIMAL FRACTIONS 101 7. If the law of problem 5 holds in 444.444, the unit of the let 4 to the right of the point equals what part of the unit of the Ist 4 to the left of the point? The unit of the 2d 4 to the right equals what part of the unit of the 1st 4 to the right? 8. In any number what part of the unit of the 1st place to the left of the point equals the unit of the 1st place to the right? of the 2d place to the right? of the 3d? of the 4th? Definition. — The unit of the 1st place, or digit, to the right is called the tenth; of the 2d place, or digit, the hundredth; of the 3d, the thousandth; of the 4th, the ten-thousandth and so on. §120. Numeration of Decimals. 1. The number 444.444 might be read, "4 hundreds, 4 tens, 4 units and 4 tenths, 4 hundredths, 4 thousandths"; but it is simpler to read it, **Four hundred forty-four a7id four hundred forty-four thousandths.^^ Bead 234.234 both ways. Which is the shorter? 2. The latter mode of reading ia the one used in practice. It maybe stated thus: Read the integral (whole) part of the number, then read the part of the number to the right of the point just as though it were a whole number standing to the left of the point, then pronounce the name of the unit of the last digit on the extreme right. The word '*and" must be pronounced only at the decimal point. Read 675.328; 236.89; 7.65; 43.6587. 3. Read the following (1) .1; .01; .001; .0101; 1.1; 10.1; (2) .6; 6.06; 66.6; .0606; 600.06; 6000.006; (3) .60; 7.070; 8.50; .0600; 87.087; 10.0008. 4. What is the ratio of 6 to .5; of .5 to .05; of .5 to .005; of 55 to 5.5; of 55 to .55? 5. Find the ratio of 875 to 87.5; of 87.5 to 8.75; of 875 to 8.75; of 875 to .875. 6. How is a number affected by moving the decimal point 1 place toward the left (see problems 4 and 5)? 2 places? 3 places? 6 places? 7. What is a quick way of dividing any number by 10? 100? 1000? 192 RATIONAL GRAMMAR SCHOOL ARITHMETIC 8. Express the ratio of 2.358 to 23.58; to 235.8; to 2358; to 23,580; to .2358; to .02358. Query. — Where is the point supposed to be in 2358? When the decimal point is not written, where is it supposed to be? 9. How is a number changed by moving the decimal point 1 place to the right? 2 places? 3 places? 6 places? 10. Make a rule for quickly multiplying any number by 10; by 100; by 1000; by 1 followed by any number of zeros. 11. Referring to problem 4, can you tell what effect is produced in such a number as .683 or .492 by writing a zero between the decimal point and its first digit? 2 zeros? any number of zeros? 12. Compare .5 with each of these numbers: .50; .500; .50000. What is the effect of writing any number of zeros to the right of the last digit of a decimal? Definitions. — A decimal fraction, or decimal, is a fraction whose denominator is 10, 100, 1000, or some power of ten, in which the denom- inator is not written but is indicated by the position of the decimal point. A power of 10 is a number obtained by using 10 as a factor any number of times. §121. To Seduce a Decimal to a Common Fraction. The fractions we have studied whose denominators are actually written, are called Common Fractions. 1. Express the following decimals as common fractions: .5; .50; .500; .4; .40; .400; .6; .60; .25; .250; .125; .375; .625; .875. 2. Reduce the results of problem 1 to their lowest terms. 3. Write these mixed decimals as improper fractions : 1.5; 2.50; 2.75; 10.4; 12.5; 6.75; 18.25. 4. Eeduce these improper fractions to their lowest terms. 5. After dropping the decimal point from the following decimals, what numbers must be written beneath them to express them as common fractions : .6; .67; .625; .875; .1275; 12.75; 25.786; 33.333; 6.6666? 6. Make a rule for expressing any decimal as a common frac- tion in its lowest terms. DECIMAL FRACTIONS 193 7. How may 2| tenths be expressed as a decimal? 62^ han- dredths? 33i hundredths? 14 and 666^ thousandths? A pure decimal is a decimal whose value is less than 1 ; as, 38 thousandths, QQ^ hundredths. A mixed decimal is a decimal whose value is greater than 1: as, 3.58 or 2.87|. 8. Read and give the meaning of these mixed decimals: .12i; 3.33i; 18.66f; 2.1^; .Of; .04f ; .OOf; 36.000|. Note. — Numbers expressed in both decimals and common fractions are called complex decimals. A simple decimal is expressed without the use of common fractions. To reduce such an expression as 3.444 to a mixed number pro- ceed thus: Or. .h„, 8.44, = 8M= 8;g= a|g j;_,?». 9. Beduce the decimals of problem 8 to mixed numbers or common fractions. 10. Make a rule for expressing any mixed decimal as a common fraction, or a mixed number. Principle I. — Any decimal may be expressed as a common fraction in its lowest terms or as a mixed nuinher^ hy dropping the decimal pointy writing the denominator^ and reducing the resulting common fraction to its lowest terms. §122. Bain and Snowfall (addition). Definition. — 1 in. of rainfall means a fall of 1 cu. in. of water on each square inch of surface of the ground. (Review §45, pp. 56-7). 1. The following quantities of rain fell from week to week during May, 1902, in Chicago; find the total rainfall for the month: First week, 1.09 in.; second week, 1.11 in.; third week, .45 in., and fourth week, 2.43 inches. Solution.— CtoNVBNiBNT Form EXPLANATION.— It is convenient to write the 1.09 m. addends in a column so that the units digits are all Ik ^"* ^^ ***® same vertical column. Then begin on the 04Q •^* right and add as with whole numbers. In the sum 2.43 in. the point should stand directly under the points in Ans, 5.08 in. the addends. Definition. — Finding the sum of decimal numbers is called addition of decimals. 194 RATIOKAL GRAMMAR SCHOOL ARITHMETIC 2. The following numbers denote the monthly rain or snow- fall for 1902; what was the total rain or snowfall for the year? .66 1.53 4.16 2.26 5.08 6.45 4.25 1.44 483 1.45 2.03 1.90 3. Without rewriting the numbers, find the total yearly rain or snowfall for the years 1891-1901 from these recorded data: Jan. FSB. Mar. Apb. May Junb July Aug. Sbpt. Ocrr. Nov. Dec. Total 1.99 1.95 2.13 2.14 2.09 2.42 2.47 4.52 .82 .36 3.83 1.32 1.99 1.57 2.21 2.17 6.77 10.58 2.23 1.85 1.34 1.54 2.68 1.63 2.08 2.44 1.69 416 1.93 8.59 3.08 .18 1.98 1.75 2 45 2.14 1.55 2.13 2.66 2.65 3.35 1.96 .60 .60 8.28 .85 1.18 1.66 2.15 1.60 1.32 36 1.99 1.79 2.43 6.49 1.89 .51 5.60 6.76 1.12 3.48 1.26 2.79 416 2.82 3.61 3.52 6.70 1.36 2.16 .16 4.53 2.22 3.56 2.23 .84 3.60 1.47 1.70 .84 .18 3.06 1.62 3.54 2.59 460 .76 2.23 5.30 1.94 3.03 3.16 3.26 2.25 1.11 .58 1.60 2.11 .14 4.35 2.71 6.66 .91 2.39 2.09 1.14 6.81 1.21 3.52 1.58 1.02 3.59 2.06 4.64 4.24 1.56 1.35 3.30 .58 1.15 2.05 3.38 .33 2.18 2.42 4.25 2.00 2.92 1.29 .85 1.70 4. Foot and average the vertical columns and tell what the footings and averages mean. §123. Other Applications. 1. A coal dealer received 8 carloads of coal of the following tonnages: 24.6, 28.785, 31.25, 24.95, 31.8, 25.125, 28, and 29.25. What was the total tonnage (number of tons)? 2. A farm was divided by its owner iuto lots of the following acreages (number of acres) : 32.87^, 7.12^, 68.33^, 11.66|, 16.28t, and 21.13f . What was the total area of the farm? 3. The following numbers represent in thousands of feet a lumber dealer's sales in 1 da. : 6.865, 24.245, 16.398. 12.28, 18.2, 6.395, 24, and 18.967. What were the total sales for the day? 4. A quantity of soil contained .125 lb. gravel; .268 lb. coarse sand; .175 lb. fine sand; .037^ lb. organic matter; .214f lb. clay, and .275 lb. water. What was the total weight of the soil? 6. During 8 hr. a freight train made the following mileages: 32.15, 28.375, 15.687, 20.2, 15.63, 17.5, 8.95, and 21.3. How far did the train run during the 8 hours? 6. Following are the weights in grains of the U. S. coins: If-piece, 48; 5ji^-piece, 73.166f ; dime, 38.583^; quarter dollar, DECIMAL FRACTIONS 196 96.45; half dollar, 192.9; dollar, 412^; quarter eagle, 64J^; half eagle, 129; eagle, 258; double eagle, 516. Find the total weight of all, §124. Nature Study — (subtraction). 1. 54 en. in. of soil in its natural state weighed 1.94 lb. After being thoroughly dried it weighed 1.459 lb. How much moisture passed off in drying? Explanation.— We liave seen that 1.94 may be written 1.940. For convenience write the numbers so that units digits stand in the same column. Be- ginning on the right subtract as though the numbers were whole numbers. In the result the point should stand directly under the points in the minuend and subtrahend. Solution.— Convenient Form 1.940 lb. 1.459 lb. Ans. .481 lb. DSFiNrnoN. — ^Finding the difference of decimal numbers is called subtraction of deeimah. 2. After drying, the same soil occupied only 37.125 cu. in. How much did it shrink in bulk in drying? 3. 100 green oak leaves weighed .22 lb. After thorough drying they weighed .087 lb. What was the weight of water contained in the green Iqaves? 4. The organic matter in the leaves was then driven off by burning the dry leaves. The ash weighed .0053 lb. How much organic matter did the 100 leaves contain? 5. The corresponding numbers for 100 green elm leaves were : Weight of green leaves, .132 lb. ; weight of dry leaves, .0345 lb. ; weight of ash, .0035 lb. Answer questions like 3 and 4 for these leaves. 6. Answer similar questions for these leaves: Kind or Leaves Weight in Pounds Fresh, Gbeen Dbt Ash 50 Poplar leaves 35 Compound ash leaves. .132 .099 .043 .044 .0043 .0026 7. 1.135 lb. of dry beans, soaked for 24 hr., weighed 2.212 lb. What was the amount of water taken up by the beans? 19G RATIONAL GRAMMAR SCHOOL ARITHMETIC 8. The dry beans occapied 34.875 en. in. and the soaKed beans 85.5 ca. in. How much did the beans increase in bulk? §125. Stature and Weight of Persons. The following table contains the average heights and weights of boys, girls, men, and women for the ages indicated by the numbers in the first column. Heights are given in feet and weights in pounds. Heights Growth in Height WmOHTS GROWTH ni Weight Age Yr. Difl. Dlff. 2 Males Females Males Females Males Females Males Females 2.60 2.56 25.01 23.58 . 4 8.04 3.00 31.38 28.67 6 3.44 3.38 38.80 35.29 9 4.00 3.92 49.95 47.10 11 4.36 426 59.77 56.57 13 472 460 75.81 72.65 15 5.07 492 96.40 89.04 17 5 36 5.10 116.56 10443 18 5.44 5.13 127.59 112.55 20 5.49 5.16 132.46 115.30 30 5.52 5.18 140.38 119.82 40 5.52 5.18 140.42 121.81 50 5.49 5.04 139.96 123.86 60 5.38 497 136.07 119.76 70 5.32 497 131.27 113.60 80 5.29 494 127.54 108.80 90 5.29 494 127.64' 108.81 1. Fill out on a separate slip the columns headed Difference (Diff.); the first by subtracting the height of the females from that of the males of the same age, and the second by subtracting the weights of females from those of males of the same age. Sab- tract without rewriting the numbers. 2. There are 4 vacant columns headed Growth; two for growth in height and two for growth in weight. Fill out on a separate slip columns like the first of these from the column of heights of males by subtracting each number of this column from the one DECIMAL FRACTIONS 197 next below it. Tell what the difference means. Fill out a column like the second similarly from the column of heights of females. 3. In a similar way fill out the last two columns from the columns of weights of males and weights of females. Tell what these differences mean. 4. At what age are boys growing most rapidly in height? in weight? At what age do men begin to decrease in height? in weight? 5. Answer questions similar to 4 for females. 6. Compare your own height and weight with the numbers of this table, for your age. Note. — ^If your age is not in the table, it will be between two ages given there. Use the mean of the numbers for these two ages for your comparison.. 7. The following table contains the height in feet of children of Manchester and of Stockport, (1) who are working in factories, and (2) who are not working in factories. Without rewriting the numbers, fill out the vacant columns of differences between the heights of the two classes for both boys and girls. What effect, if any, of such work can you detect (find out) on the growth of the boy or girl? Note. — When the boy or girl not working in factories is taller than the one forking in factories, mark the difference with a plus {-{-) sign before it. In the opposite case mark the difference with the minus ( - ) sign before it. Boys GTRT<8 Agbs DIFT. Dirr. Working In Factories Not Working Working in Factories Not Working in Factories in Factories 9 years.. 4.009 4.045 8.996 4.036 10 - 4.167 4.219 4.134 4.114 11 •• 4.272 4.252 4.261 4341 12 •' 4.446 4.413 4.475 . 4.472 13 •* ' 4.537 4.580 4.636 4.590 14 " 4.715 4.725 4.813 4.852 15 " 4.971 4.836 4.875 4.928 16 " 5.134 5.266 4.990 5.003 17 " 5 228 5.338 5.039 5.059 18 " 5.276 5.825 5.226 5397 198 RATIONAL GRAMMAR SCHOOL ARITHMETIC §126. Pointing the Prodnet of Deeimals— (multiplication). ORAL WORK 1. What is the relation between the following pairs of num- bers? (1) 25 and 2.5 (4) 1.28 and 12.8 (7) 2847 and 28.4? (2) 25 and .25 (5) 1.28 and 128 (8) 284.7 and 2.847 (3) 75 and 7.5 (6) 47.8 and 4.78 (9) 28.47 and .2847 2. The product 37 x 25 equals how many times the product 37x2.5? 3. The product 684 x 7.5 equals what part of the product 684 X 75? 4. What is the product 684 x 75? What, then, is the product 684 X 7.5? What is the product 684 x .75? 684 x .075? 68.4 x.075? 6.84 X. 075? .684 x.075? Definition.— By the number of decimal places of a number is meant the number of digits (zero included) on the right of the decimal point. Illustration. — In 2.005 there are 3 decimal place& WRITTEN WORK 1. Find the product 4862 x 784, and from it write the follow- ing products : 486.2 X 784; 48.62 x 784; 48.62 x 78.4; 4.862 x 7.84; 4862 x .784. 2. How many decimal places are there in 486.2? in 48.62? 4.862? 10.03? 3.0060? .0600? 20.0806? 3. How many decimal places are there in each of the prod- ucts of problem 1? 4. Compare the number of decimal places in each of the prod- ucts of problem 1 with the sum of the numbers of decimal places in both the multiplicand and the multiplier. What do you find? 5. Make a rule for finding how many decimal places there must be in the product of two decimals. 6. How, then, can you find where the decimal point belongs in the product of any two decimals? DECIMAL FRACTI0K8 190 Principle II. — The number of decimal places in the product equals the sum of the numbers of decimal places in the factors, 7. A field containing 38.75 A. yielded 23.9 bu. of wheat per acre ; what was the total yield? Solution. — Since each acre yielded 23.9 bu., 88.75 A. must yield 88.75 X 23.9 bushels. CONYENUBirT FORM EXPLANATION. — ^m ^«I5 239 X 38.75 = how many times 23.9 X 38.75? 23.9 239 239X3875= ** *' 239x38.75? 239 X 3875 = " " 23.9 X 38.75? What is irftto of 926,126? Does the rule of problem 5 hold true here? 926.125 926125 Ana. 926.125 bu. 8. A steer weighing 16.22 cwt. sold at $8.85 per cwt. ; what price did he bring? 9. A passenger train ran for 12.27 hr. at the rate of G5.75 mi. per hour; how far did it run during the time? §127. Force Needed to Draw Loads on Eoad Wagon. 1. To draw a load in a road wagon at a slow walk over hard, level country roads a horizontal pull of .075 of the total weight of the wagon and load is required. What horizontal pulls will be required to draw the following : 1195 1195 1678 1673 1912 1912 717 717 Weight or Wagon Weight or Load Total Weight Horizontal Pull in L.b. (1) 1068 lb. 2168 lb. .... .... (2) 969.5 2408.3 .... . • • • (3) 1580.75 . 3675.6 .... .... (4) 2368 3890 .... .... 2. Over fresh earth .125 of the total load as a horizontal pull will draw it on a road wagon. Fill out the vacant columns, for these conditions : WlIOBT OF WAQON Weight OF Load Total Weight Horizontal Pull in Lb. (1) 980 lb. 1260 lb. .... .... (2) 940.8 768.78 .... .... (3) 1164 3890.85 .... .... (4) 1675 4060.75 .... 200 RATIONAL GEAMMAB SCHOOL ARITHMETIC 3. Over loose sand .258 of the total load will draw it. Find the forces needed to draw these loads : Wbight of Wagon WBIGHT OF LOAD Total Weight HOSIZONTAI. PULIi IN Lb. (1) 375 1b. 686.9 lb. .... .... (2) 1860 2586.38 .... • . ■ • (3) 2800 3869.25 «... • ■ . . 4. On good, broken stone pavement the pull is about .0285 of the total load; on wood pavement it is .019 of the total load; and on Macadam pavement it is .0333 of the load. Find the pull in pounds needed for each of these three kinds of pavement for the following: WmOHT OF Wagon Weight of Load Total Weight Horizontal PULL IN Lb. (1) 4060 lb. 3795.65 lb. .... .... (2) 5190 6340.86 .... .... (3) 4960 7640.65 .... • . • • §128. DiviBion by an Integer. 1. I paid $941.25 for 15 A: of land; what was the price per acre? Explanation. — Compare the steps in the work with decimals with the corre- sponding steps in the work with Integers. How do the decimal points stand through the problem? How does the point stand in the quotient? How may the dividend be found from the divisor and the Quotient? How, then, may you check division? Ana. $62.75. 2. I paid $2823.75 for 45 A. of land; what was the price paid per acre? Convenient Form With Decimals With Integers 62.75 6275 15)941.26 90 15)94125 90 41 30 30 11.2 10.5 112 105 .75 .76 75 75 DECIMAL FRACTIONS 201 3. The 36 members of a society were assessed (made to pay) equally to meet a debt of $1341 against the society. What was the amount of the assessment against each member? Solution. — 87.25 36)1341.00 108 "iii 252 9.0 7.2 lio 1.80 Notice particularly how by writing zeros after the deci- mal point the quotient may be carried out in a decimal form. What effect does writing zeros after the point have on a number? Ans. $37.25. 4. The number of states and territories and the total areas in square miles of the land surface of the principal geographic divisions of continental United States according to the Twelfth Census, are given in the following table. Find the average land surface of a state for each division and for the whole United States. Carry the division to three decimal places.* Division Number Land Subfacb Average North Atlantic 9 9 12 9 11 162,103 168.620 753.550 610,215 1,175,742 . South Atlantic North Central South Central Western Continental United States 6. The 10 loads given in pounds in column 2 of the following table required the number of pounds of force given in the third column, to draw them on a common road wagon over a good, level country road. For example, 1,400 lb. required 98 lb. to pull it; «If the next decimal place after the last one required is less than 5, write the given quotient as the required decimal. If it is greater than 5, add 1 to the last figure of tbe decimal required. This rule applies to all cases of division of decimals. 202 RATIONAL GRAMMAR SCHOOL ARITHMETIC 1616 lb. required 112 lb., and so on. Divide each load by the number of pounds of force needed to draw it, and put the quotient to two decimal places in the column headed "Ratio." •BSIMENT LOAD Puu. 1 1400 98 2 1616 112 3 1825 126 4 2236 155 5 2440 170 6 2650 185 7 2863 198 8 3072 216 9 3281 229 10 3662 244 Ratio §129. DiviBion by a Decimal. 1. What is Vt ot 55? 2. When no decimal point is written with a number where is it understood to be? 3. Where is the decimal point in 55? 4. How is a number changed by moving its decimal point 1 place toward the left? 3 places toward the left? 1 place toward the right? 2 places toward the right? 5. 55 equals how many times 5.5? .55? .055? 6. If 55 + 11 = 5, to what is 5.5 ^ 11 equal? ,55 ^ 11? 7. Name these quotients : 250 -s- 25 ; 25 -f- 25 ; 2.5 -i- 25 ; .25 + 25. 8. Name these quotients: 250 -*- 2.5; 25 -*- 2.5; 25 -*- .25; 250-^^.25. 9. 2176 + 32 = 68; to what number is x equal in each of the following equations : (1) 217.6 ^32 = a:; (4) 217.6 + 3.2 = a?; (7) 2176 ^ 3.2 = a;; (2) 21.76-^32 = 2;; (5) 217.6 -^ .32 = a;; (8) 21.76+ .32 = a;; (3) 2.176 + 32 = a;; (6) 21.76 + 3.2 = a;; (9) 2.176 + .032 = a:? 10. In each case of problem 9 compare the number of decimal places in the quotient with the number of decimal places in the dividend minus the number in the divisor. DECIMAL FK ACTIONS 203 11. 101,388-426 = 238; without dividing,write out the numbers to which X is equal in the following equations: (1) 10138.8-*-426=a;; (4) 101388 H-42.6=a;; (7) 101.3884- 42.6=2;; (2) 1013.88-^426=2;; (5) 101388 -^4.26=2;; (8) 10.1388-*- .426=2;; (3) 1.01388-^426=2;; (6) 1013. 88 -^4. 26= a;; (9) 1.01 388 4-. 0426 =2;. 12. From the results of problems 10 and 11 make a rule for finding the number of decimal places in the quotient. Principle III. — The number of decimal places in the quotient equals the number of decimal places in the dividend minus the num- ber of decimal places in the divisor. §130. Problems. 1. In 1891 the total imports of tea into U. S. were 82,395,924 lb. and of coffee 611,041,459 lb. If the average cost of tea was 37^ per lb., and of coffee 18^, what was the total cost of both? 2. On Dec. 6, 1901, a carload of 34 Angus show cattle of weights given in table annexed sold at the prices per cwt. set beside the weights. What was the total weight of the carload? the average weight of the animals? 3. What was the average price per hundredweight? 4. For how much did the entire load sell? 5. What was the average price per head which the owner received for the carload? 6. The previous year on the same occasion, a carload of 26 prize-winning Angus cattle, averaging 1492 lb., sold at $15.50 per hundredweight. What aver- age price did the cattle bring the owner? 7. How much did he receive for the carload, problem 6? No. Weight Prigb 1 1503 $ 9.00 2 1622 8.85 3 1606 8.60 4 1524 8.50 5 1524 8.50 6 1504 8.10 7 936 8.70 8 1327 6.85 9 1273 8.05 10 1130 8.75 11 1326 7.65 12 1141 8.50 13 1318 8.75 14 1327 8.70 15 1190 7.70 16 1446 7.85 17 1449 7.85 18 1542 7.70 19 980 6.80 20 1460 8.10 21 852 8.30 22 1376 8.30 23 1540 25.00 24 1631 9.30 25 1468 8.65 26 1297 8.50 27 1529 8.20 28 1100 7.60 29 1073 7.60 30 1110 8.10 31 1095 8.15 32 1456 8.75 33 1298 8.00 34 2130 10.75 204 BATIOKAL QRAMMAR SCHOOL ARITHMETIC 8. A dairyman finds that during November one of his cows furnished this record : Morning Milkings Evening Miuungs Ist week 47.2 lb. 68.6 " 58.8 *' 62.7 ** 17.8 *' 40.4 lb. 48.0 '• 49.8 " 47.3 - 16.4 " 2d " 3d " 4th " 29th and 30th davs What was the total number of pounds of milk given by this cow during the month at the morning milkings? at the evening milkings? at both milkings? 9. If milk was worth 6.25^ a quart (8.6 lb. per gallon), what was the milk of this one cow worth to the owner during November? §131. Batio of Circumference of Circle to Diameter. 1. The distance around the rung of a chair was measured and found to be 3.625 in.; the diameter was found to be 1.153 in. Divide the distance around the rung by the diameter and find the quotient to 3 decimal places. Definition. — The distance around a circle is the circumference of the circle. 2. The circumference of a circular rod, 1.875 in. in diameter, was measured and found to be 5.884 in. Find the ratio to 3 decimal places, of the circumference to the diameter. 3. Measure the diameters and the circumferences of any circles in your schoolroom and find to 3 decimal places the ratio of their circumferences to their diameters. If no circles are at hand use the measures of this table : Object CIRCUMPBRBNOB DIAMBTBR Ratio Ink bottle 5.5 in. 6.154 26.538 57.080 71.458 18.125 1.75 in. 1.953 8.444 18.062 22.750 5.675 Tin box Globe of lamp Terrestrial globe Barrel-head Iron ring Average ...... DECIMAL FRACTIONS 205 4. Measure the circumferences and the diameters of the fol- lowing objects and find the ratios to three decimal places. If you are unable to make your own measures use those of the table: Object ClBCUMrERENCB Diameter Ratio Bicycle wheel Bicycle wheel Front carriage wheel Rear Locomotive driver . . Averasre 81.177 88.055 151.189 176.333 174.762 26.025 ' 28.012 48.125 56.125 65.625 5. Find the ratio of the circumferences to the diameters of the coins of problem 27, p. 108. 6. It is proved in Geometry that the ratio of the circumference of any circle to its diameter is about 3^^, or more accurately 3.1416. If then, the diameter of a circle is known, how may the circum- ference be found without measurement? 7. Find the average of all the values of the ratios found in prob- lems 1 to 4 and compare this last average with 3^; with 3.1416. What is the difference in each case? Note. — For most practical purposes this ratio, denoted by the Greek letter ir, and called pi, may be taken as 3k For greater accuracy use ir = 3.1416. §132. Original Problems. Make and solve problems based on the following facts : 1. 1 cu. ft. is about .8 of a bushel of small grain. A grain bin is 8' X 12' X 22'. 2. 1 bn. of ear corn is about 2.25 cu. ft. A wagon box is 2.G7' X 2.85' X 9.6'. 3. Well settled timothy hay runs about 355.25 cu. ft. to the ton. A hay shed is 24' x 40' and is filled with hay to a height of 18.5'. 4. Loose timothy hay runs about 460 cu. ft. to the ton. A load of hay is 8' x 18' x 22.7'. 5. Stove coal runs about 35.1 cu. ft. to the ton. A coal bin is 6' X 12.5' X 15.35'. 6. 1 perch of stone is 24.75 cu. ft. A stone wall is 2.75' x 4.385' X 126.8'. 206 RATIOKAL GRAMMAR SCHOOL ARITHMETIC 7. A man walks about 3.5 mi. per hour. It is 85 mi. from Chicago to Milwaukee. 8. A horse trots about 7.5 mi. per hour. 9. A horse runs about 18 mi. per hour for short distances. 10. A steamboat runs 18 mi. per hour. 11. A slow river flows 3 mi. per hour. 12. A rapid river flows 7 mi. per hour. ' 13. A crow flies 25 mi. per hour; a falcon, 75 mi.; a wild duck, 90 mi. ; a sparrow, 92 mi. ; and a hawk, 150 mi. per hour. 14. A carrier pigeon flies 80 mi. per hour for long distances. 15. Sound travels through air 1134 ft. per second; through water, 5000 ft. per second; and through iron or steel, 17,000 ft. per second. 16. A rifle ball travels 1460 ft. per second at starting; and a 20-lb. cannon ball, 16,000 ft. per second at starting. 17. Light travels 186,600 mi. per second; electricity, 288,000 mi. per second. The sun is 93,000,000 mi. from the earth ; the moon, 240,000 miles. 18. One horse-power raises 33,000 lb. through a height of 1 ft. in 1 minute. 19. The equatorial diameter of the earth is 7925.6 mi.; the polar diameter, 7899.1 miles. §133. Physical MeasurementB. The following table gives, for men, of ages from 18 to 26 years, the average lung capacity in cubic inches, the height in inches, and the weight in pounds. AGE Lung Capacity Height Weight 18 251.4 68.2 134.25 19 251.8 68.2 135.40 20 258.2 68.1 138.65 21 260.4 68.1 140.60 22 264.8 68.2 141.15 23 263.7 68.1 138.60 24 267.1 68.2 143.90 25 267.2 68.2 143.15 26 267.1 68.9 142.80 DECIMAL FRACTIONS 207 1. Find the number of cubic inches of lung capacity per inch of height, for each age, by computing (finding) the ratio to 2 deci- mal places of each number of column 2 to the corresponding num- ber of column 3. Is this ratio the same for all ages? 2. Similarly find the number of cubic inches, to 2 decimal places, of lung capacity per pound of weight, for each age. 3. For each age find the number of pounds of weight per inch of height to 2 decimal places. Are these numbers the same for all ages? §134. Specific Gravity. The specific gravity of any solid or liquid substance is the ratio of its weight to the weight of an equal bulk of water. The weight of a cubic foot of water is G2.5 pounds. 1. The following table contains the weight in pounds of 1 cu. ft. of the substances mentioned. Find to 3 decimal places the specific gravities of these substances : Metal Aluminum. Zinc Cast iron . . Tin Wr't iron. . Steel Brass Copper Silver Lead Gold Platinum . 166.5 436.5 450.0 458.3 480.0 490.0 628.8 552.0 655.1 709.4 1200.9 1847.0 mo Wood Cork Spruce . . Pine (yellow) Cedar Pine (white) Walnut Maple Ash Beech Oak Ebony Lignum vitae (1)0 15.00 3L25 84.60 35.06 28.00 41.90 46.88 52.80 53.25 65.00 76.00 83.30 Liquid Alcohol . . . Turpentine Petroleum Olive oil . . Linseed oil Sea water . Milk Acetic acid Muriatic acid Nitric acid. Sulphuric acid Mercury . o o . |2 50.0 54.4 55.7 57.0 59.5 64.1 64.5 66.5 75.0 95.0 115.1 880.0 mo 2. The specific gravity of any gas or vapor, is the ratio of its weight to the weight of an equal volume of air, the gas or vapor and the air being at the same temperature and under the same 208 RATIONAL GRAMMAR 8CH00L ARITHMETIC pressure. Find to 3 decimal places the specific gravities of the gases and vapors in the following table : Gas Weight, in Pounds, or 1 Cubic Foot Specific Gravity Hvdrofiren .00559 .00727 .00815 .03790 .07810 .07860 .08073 .08925 .12344 .19700 Smoke (wood) Smoke fsof t coal) Steam at 212° F Carbonic oxide Nitro&ren Air Oxygen Carbonic acid Chlorine 3. The following familiar substances may be compared with water as to weight. Find their specific gravities to 2 decimal places : Substance Weight, in Pounds, OF 1 Cubic Foot Specific Gravity Glass (average) 175.8 174.5 169.2 166.4 158.2 133.4 124.5 121.8 118.3 113.9 Chalk Marble Granite Stone (common) Salt " Soil " Clay Brick Sand §135. To Eeduce a Common Fraction to a Decimal. 1. Express $f decimally. Solution .375 8)3.000 2.4 "jftO .66 .040 .040 Explanation. — Annex zeros to the right of the decimal point after the numerator, and then divide by the denominator. ^Ins. $i=$.875 DECIMAL FBACTI0N8 209 2. Express the following common fractions decimally (to 3 decimal places) : il il i> f; fj TIT 5 /?> H« 3. Express the following mixed numbers decimally (to 3 places) : H; 3i; 2i; 6f; 13^^; 18 Jf 4. Express the following decimally to 5 decimal places: ^5 tVj tt> ifV; H> A- Such decimals as these fractions give rise to, that do not ter- minate, are called non'terminating decimals. 5. Express the following numbers decimally to 6 decimal places : i;l; \\ f; i; I; i; i*t; A; tV; 4?; H; If DEFmrriON. — Such non-terminating decimals as these that repeat the same digit or group of digits indefinitely, are called repetends, or circula- ting deciTnals, or circulates. 6. Express the values of these numbers by the use of integers and decimals only : 1.3i; 16.87i; 5.281; 17.9|; 20.0|; l.OOf; .OOJ ; .OJ; .000,^; 30.060f, 7. Express the following fractions decimally to 4 places: ft, t^, tNf, TT> ^^, ^y 7 y, 12.17' 12f 8. A man sold 37i A. of his farm of 79 J A. ; how many hundredths of his farm did he sell? 9. A man owned a piece of city land 450' square. A strip 37 J' wide was cut from each of its four sides for streets. How many hundredths of the square were cut away? 10. How many hundredths of the area of a page of this book are in the margins? (Measure to the nearest 16th of an inch). 11. How many hundredths of the area of the surface of your desk is the area of the surface of your book? 12. How many hundredths of the area of the surface of the floor is the area of the surface of your desk? 210 RATIONAL GRAMMAR SCHOOL ARITHMETIC B. Aroa of a Circle. Draw a circle and diameter AB (see Fig. 132). With an opening of the compasses equal to the radius of the circle and with Figure 132 A as center draw short arcs at D and G^ also, with same radius and with B as center, draw short arcs at E and F. Draw radii OD^ OE, OF, and OQ. Bisect angle AOD, as in Problem II, p. 176, and draw 08, With an opening of the compasses equal to the distance between D and H and with center D draw an arc at J\ with center B draw arcs at K and L ; also, with center O draw arcs at if and N. Draw radii OJ, OK, OL, OM, and ON. Cut the circle into the twelve equal sectors thus formed and place these sectors as shown in the second part of the figure. How long is the base AB oi the approximate parallelogram thus formed? How wide is the approximate parallelogram? If each of the twelve sectors were split into halves and the resulting twenty-four sectors were fitted together as are the twelve sectors, would the wavy base line become more nearly straight? How long and how wide would the new approximate parallelogram be? What would be its area? Having the circumference and the radius of a circle how can you find the area of the circle? Having the radius of any circle how can you find the area of the circle (See §131)? PROBLEMS In problems 1 to 3, inclusive, use «• = 3if. Let r denote the length of the radius of a circle, let c denote its circumference, and Ay its area. 1. Find the areas, -4, of these circles (1) r = 12.5', c = 78.54'; (3) r = 20", c = 125.664"; (2) r = 6.25', c = 39.27'; (4) r = 60", c = 377.143". DECIMAL FRACTIONS 211 2. The diameter of a drum-head is 2.5' and the circumference 18 7.854\ how many square inches of skin are in the two heads? 3. The diameter of a circular window is 18.5" and its circum- ference is 58.12", what is its area? In the following problems v is taken as 3.1416. Results should be correct to 4 decimal places. 4. The wind is blowing squarely against a circular signboard, whose diameter is 32.75 ft., with a pressure of 25.5 lb. to the square foot. Find the total wind pressure against the board. 5. The circumference of a cylindrical chair rung is 5.5"; find the diameter and area of the right section of the rung. Note. — A right section is the section that would be made by sawing the rung square across. 6. Steam passes from the boiler of an engine through the ^ passage, P, into the cylinder, and pushes jy^ vk against the circular piston, C^ with a pres- Y^\ — Ir , "7*^ sure of 65 lb. to the square inch. If the diameter of the circular head, C, is 12.5", what is the total pressure against the left side FiouiiE 133 ^f ^1^3 pjg^Q^p 7. Answer similar qnestions for pistons having these diam- eters with steam pressures per square inch as indicated: DIAMBTEB OF PRESSURE PER PISTON Square Inch (1) 10.85" 86 (2) 22.35" 120 (3) 14.85" 180 (4) 16.45" 175 (5) 20.75" 125 (6) 15.85" 160 (7) 21.35" 205 8. Find the area, ^, of a circle whose radius is r ft. long and whose circumference is c ft. long. 9. Find the circumference, c, of a circle whose radius is r rods long. Find the area. 10. A cow is tied to the corner of a corn crib, 18' x 18', with a rope 18' long. Over how many square feet can she graze? 212 RATIONAL GRAMMAR SCHOOL ARITHMETIC §137. Exercises for Practice. 1. Add the following : [I) 37.8 (2) 248.001 (3) .96 (4) 7.68 (5) 862.01 (6) 328. 8.65 16.25 3.004 86.043 2.8106 .032 12.08 9.018 10. 190.608 843.7 7.001} .75 24. 7. 9. 1200. 29.080i 16. 3. .206 .9 .16 109. 14.01 .03 200. 8.3 9. 9000. .6 .4 860.03 60.5 .0068 .08 10. 17.003 .2 3.7 .0309 .0007 2. Subtract the following : (1) 84.96 (2) 98.01 (3) 100.01 (4) 106.038 (5) 67.083 (6) 800.00 6.38 1.97 87.08 69.879 29.187 98.68 (7)810.02 (8)3.786(9)26. (10)3. (11)68. (12)1.0101 86.983 .989 .748 2.983 9.8789 .9008 3. Find the following products: (1) 89.3 X 42.1 (7) 8.08 x .008 (13) 31.416 x 2.518 (2) 68.01 X 9.82 (8) 16.8 x .072 (14) 314.16 x 251.8 (3) 86.3 X 8.761 (9) 9.001 x 1.801 (15) 785.4 x 2.781 (4) 100.001 X 86.8 (10) 1.68 x 7.2 (16) .7854 x .2781 (5) 76.9 X .93 (11) 168 x .072 (17) 3 1416 x 16.84 (6) 7.69 X .093 (12) 3.1416 x 25.18 (18) 314.16 x .1684 4. Find the following quotients : (1) 120.716 -^ 26.82 (5) 109.624 -^ 3.86 (9) 443.52 ^ .96 (2) 123.1498 -^ 26.82 (6) 548.12 ^ 14.2 (10) 14.784 -^ .032 (3) 276.496 -^ 3.142 (7) 1774.08 -*- .384 (11) .3696 ^ .0008 (4) 268.928 ^ 8.4 (8) 8.8704 -^ 192 (12) 9.24 -^ .00004 5. Find the following quotients to 3 decimal places: (1)8^.3 (7)2^3 (13)4-^.011 (19)1.74^9.09 (2) 031 -^ .072 (8) 4 -*- 9 (14) 2.5 ^ .033 (20) .77 -*- 1.21 (3) 9 ^ .007 (9) 44 -^ .09 (15) 260 -^ 3.3 (21) 1.111 4..099 (4) 11 -^ .0009 (10) 13 H- 3 (16) 2.6 -^ 33 (22) 60.6 -^ 1.818 (5) 12 H- 36 (11) 7.7 ^ 9 (17) 11.5 ^ 3.33 (23) .625 ^ 3.36 (C) 1 -^ 3 (12) 123 ^ 9.99 (18) 10.4 ^ .909 (24) .1 -*- .09 COMlPOUND DENOMINATE NUMBERS 213 COMPOUND DENOMINATE NTJMBEBS §138. Definitions. A denominate number is a number whose unit is concrete; as,- 13 mi., 8 hr., 40 A., $125, etc. A concrete unit is a unit having a specific name; as, 1 yd., 1 lb., $1, 1 hat, 1 horse, etc. A compound denominate number is a number expressed in two or more units of the same kind; as, 11 hr. 25 min. 15 sec; 12 gal. ■J qt. 1 pt. 3 gills. TABLES OF MEASURES Note. — Read carefully the tables that follow and fix them in mind by solving the problems beginning on page 219. §139. Measures of Value. The standards of value of the United States and of some of the European countries are here given with both their rough and their accurate equivalents in U. S. money. Rough Accurate Country Standard Symbol Equivalen't Equivalent United States Dollar $ §1. $1. Great Britain Pound (Sterling) £ $5.00 $4.8665 Germany Mark (Reichsmark) M. $ .25 $ .2385 France Franc F. $ .20 $ .193 Russia Ruble R. $ .75 | .772 Austria-Hungary Crown, or Filler C. or F. $ .20 $ .203 Italy Lira L. $ .20 $ .193 Ta^le of U. S. Money 10 mills (m.) = 1 cent (ct. or ^) 10 cents = 1 dime (d.) 10 dimes = 1 dollar 10 dollars = 1 eagle 5 dollars = | eagle 2i dollars = J eagle 20 dollars = 1 double eagle The coins of the United States are bronze, nickel, silver, and gold. Table of English Money 4 farthings (far.) = 1 penny (d.) 12 pence = 1 shilling (s.) 20 shillings = 1 pound (£) 21 shillings = 1 guinea 214 RATIONAL GRAMMAR SCHOOL ARITHMETIC Table of Monetary Units of Other Nations Germany, 1 mark (M.) = 100 pfennige (pf.) France, 1 franc (fr.) = 100 centimes (c.) Russia, 1 ruble (r.) = 100 copecks (c.) Austria-Hungary, 1 crown, or fiHer = 100 heller Italy, 1 lira =100 centessimi §140. Measures of Weight. Three systems of weight units are used in the United States, viz. : troy weight, avoirdupois weight, and apothecaries' weight. Troy weight is used in weighing gold, silver, and jewels. Avoirdupois weight is used for weighing all ordinary articles, and apothecaries' weight is used by druggists in mixing medicines. The standard of weight in the U. S. is the troy pound. The grain is the same in all three systems; the pound the same in tro} and in apothecaries' weight and different in avoirdupois. Table of Troy Weight Table of Equivalents 24 grains (gr.) = 1 pennyweight (dwt) 5760 gr. \ 20 penny weights = 1 ounce (oz.) 240 dwt. >• = 1 lb. 12 ounces = 1 pound (lb.) 12 oz. ) Table of Avoirdupois Weight 7000 grains (gr.) = 1 pound (lb.) 16 ounces = 1 pound (lb.) 100 pounds = 1 hundredweight (cwt.) 2000 pounds = 1 ton (T.) 2240 pounds = 1 long ton (L.T.) Table of Apothecaries' Weight 20 grains (gr.) = 1 scruple (3) 3 scruples = 1 dram (3) 8 drams = 1 ounce (1) 12 ounces = 1 pound (lb.) §141. Measures of Length, or Distance (Linear Measure). The standard unit for measures of length, or distance, is the yard. Table of Common Linear Measure Table of Equivalents 12 inches (in.) = 1 foot (ft.) 63360 in. ^ 3 feet =1 yard (yd.) 5280 ft. [ __ ^ yards, or 161 ft. = 1 rod (rd.) 1760 yd. [ " ^ °^^ 320 rods =1 mile (mi.) 320 rd. COMPOUND DENOMINATE NUMBERS 215 Table op Surveyors' Linear Measure 7.92 inches = llink (li.) 100 links == 1 chain (ch.) 80 chains = 1 mile (rai.) 1^142. Measures of Surface. The unit upon which surface measure is based is the square yard^ which is a square each of whose sides equals 1 yard. Table op Common Surpace Measure 144 square inches (sq. in.) = 1 square foot (sq. ft.) 9 square feet = 1 square yard (sq. yd. ) 30 J square yards = 1 square rod (sq. rd.) 160 square rods = 1 acre (A.) Table op Surveyors' Surface Measure 625 square links (sq. li. ) = 1 square rod (sq. rd. ) 16 square rods = 1 square chain (sq. ch.) 10 square chains = 1 acre (A.) 640 acres (a section) = 1 square mile (sq. mi.) 86 square miles = 1 township (Tp. ) Table of Equivalents 3686400 sq. rd. j 280400 sq. ch. [ = 1 Tp. 36 sq. mi. ) §143. Measures of Volume. Table op Cubic Measure 1728 cubic inches (cu. in.) = 1 cubic foot (ou. ft.) 27 cubic feet = 1 cubic yard (cu. yd.) Table op Equivalents 46656 cu. in. ) , , 27c«.ft. {=!««• yd. A perch of stone is a square-cornered mass, 1' x 1|' x 16^, or 24i cubic feet. Fire wood is measured by the cord. A cord of wood is a straight pile, 4' x 4' x 8', or 128 cubic feet. A cord foot is a straight pile of wood, 4' x 4' x 1'. How many cubic feet are there in a cord foot? 21G RATIONAL GRAMMAR SCHOOL ARITHMETIC §144. Measures of Capacity* Liquid measure is used in measuring liquids, the capacity of cisterns, tanks, etc. Table op Liquid Measure 4 gills (gi.) = 1 pint (pt.) 2 pints = 1 quart (qt. ) 4 quarts = 1 gallon (gal. } Note. — A liquid gallon contains 231 cubic inches. Table op Equivalents 32 gi. J , 8 pt. [ = 1 gal. 4qt. ) Dry measure is used in measuring grain, fruit, vegetables, etc. The standard of dry measure is the Winchester bushel^ which is a cylinder 18^ in. in diameter and 8 in. deep, containing 2150.42 cubic inches. Table^of Equivalents 64 pt. 32 qt. S- = 1 bu. Table of Dry Measure 2 pints (pt. ) =7= 1 quart (qt.) 8 quarts = 1 peck (pk.) 4 pecks = 1 bushel (bu. ) Note. — A dry gallon = 4 qt. or \ pk., or 268.8 cubic inches. avoirdupois pounds in a bushel prescribed by various states D4pt. \ 32 qt. [ =\ 4pk. ) COMMQUmZS, BBFlef .»....>' BeaD^ < Blue Grass Seed . Buckv'hetLt ,«i Castor Ubuqs — ^< CJcver Seed CorD on the Cob . , Cora, Sbelled.., ,. Cartimeai Dried Apples.,,.. Dried Pouches Flax Seed Bcmp Set^d .., Miilbt ..*„.. Oftts Onlana. ........... I>«i4B Potatoes .*.- Rye . , , Sweet Potatoes .. TlttitJtliy Seed .„. iTurcipB Wheat , . . „ , 50 EiO as: 54 50 48 44 nO 32 57 45 ^ < 52 50 4a 60 m 70 5G .3 47 4^' 60W 14 14 fiO 56 46 WflO 70' 70 50,50 24 24 33 39 56 56 44 44 3S33 57,67 flO 60,60 61) ?fi '56^56 M' 46 50"55i.J 45 45 1 45' t^ m\ UU6U60 m m 4& 4H 6U |GU. 62 m 5^ 60100 3S m 2E24 33 55 44 tfy ^ m m m- 5fi 47 GO .. 33 SO 56 60 «0 56 50.. 45 mm 4S i% 4e ^At«0 60 50,42,4U ,Jeo 070 56,56 441 50 60 fi6;57 60 00, 6D 6C,5e 50! 55 4&i4& 5& 60.60j6Q 48' 4e mm u 5e 6« 45 41] @oe 60t&7 COMPOUND DENOMINATE NUMBERS 217 §145. Measures of Time. The standard unit for measuring time is the mean solar day. The mean solar day is the average time interval from the instant when the sun crosses the meridian of a place (noon) to the next instant of crossing the meridian (the next noon). Table of Time Measure 60 seconds (sec.) = 1 minute (min.) 60 minutes = 1 hour (hr.) 24 hours = 1 day (da. ) 7 days = 1 week (wk.) 30 days = 1 calendar month (mo.) 12 calendar months = 1 calendar year (yr.) 365 days = 1 common year 366 days = 1 leap year 100 years = 1 century Note. --One mean solar year = 365 da. 5 hr. 48 min. 46 sec. = 365} da. (nearly). Table op Equivalents 31536000 sec. ^ 525600 min. 8760 hr. 365 da. 52 wk. 1 da 12 mo. >. = 1 common yr. According to the Julian calendar^ adopted by Julius Caesar (whence the name), every year whose date number (as 1896, 1904) is divisible by 4 must contain 366 da. and all other years, contain 365 da. Years containing 366 da. are called leap years; those containing 365 da. are called common years. According to the Gregorian calendar^ which is now used by nearly all civilized nations, every year whose date number is divisible by 4 is a leap year, unless the date number ends in two zeros (as 1600,- 1900), in which case the date number must be divisible by 400 to be a leap year. The extra day of the leap year is added to February, giving this month 29 da. in leap years. §146. Measurement by Counting. Table for Counting Table for Measuring Paper 12 things = 1 dozen (doz.) 24 sheets = 1 quire 20 things = 1 score 20 quires = 1 ream 12 dozei;! = 1 gross 2 reams = 1 bundle 12 gross = 1 great gross 5 bundles = 1 bale 218 RATIONAL GBAMMAR SCHOOL ARITHMETIC The operations to be performed in compound denominate numbers are the following: (1) To change from 'higher to lower units,, or denominations; (2) To change from lower to higher units, or denominations; (3) To add compound denominate numbers; (4) To subtract such numbers; (5) To multiply such numbers; (6) To divide such numbers. These processes will now be illustrated in order. 1. Express 16 bu. 3 pk. 3 qt. 1 pt. as pints. Convenient Form 16 bu. 3 pk. 3 qt. 1 pt. 4 = No. pk. in 1 bu. 64 pk. = No. pk. in 16 bu. 3pk. 67 pk. = No. pk. in 16 bu. 3pk. 8 = No. qt. in 1 pk. 536 qt. = No. qt. in 67 pk. 3qt. 539 qt. = No. qt. in 67 pk. 3 qt 2 = No. pt. in 1 qt. 1078 pt. = No. pt. in 539 qt. 1 pt. Am. 1079 pt. = No. pt. in 16 bu. 3 pk. 3 qt. 1 pt. 2. Express (1) 1079 pt. in higher units; (2) 2 pk. 3 qt. 1 pt. in bushels. Convenient Form (1) 2 ) 1079 pt. 8 ) 539 qt. + 1 pt. remaining 4 ) 67 pk. + 3 qt. remaining 16 bu. + 3 pk. remaining Ans, 16 bu. 3 pk. 3 qt. 1 pt. (3) 8 qt. 1 pt. = 3.5 qt. = ^ pk. = .4875 pk. 2 pk. 3 qt. 1 pt. = 2.4375 pk. = ^^^ bu. = .609375 bu. Am. COMPOUND DENOMINATE NUMBERS 219 3. The three sides of a triangular grass plot were: 8 yd. 2 ft. 10 in. ; 12 yd. 1 ft. 9 in. ; and 9 yd. 2 ft. 7 in. ; how far is it around the plot? Explanation.— First, adding the inches, Convenient Form ^^ ^^^^^^ ^g j^ = 2 ft. 2 in. Write the 2 in. 8 yd. 2 ft. 10 in. in 'inches" column, and add the 2 ft. to the 12 1 9 numbers in the '"feet" column, giving 7 ft. 9 2 7 = 2 yd. 1 ft. Write 1 ft. in '*feet" column Ans 31 vd 1 ft 2 in *^^ *^^ ? ^^' ^ ^^® numbers in **yards" col- ^ ' ' umn, giving 31 yd. 4. From a vessel containing 25 gal. 3 qt. 1 pt. 2 gi. of oil, 8 gal. 3 qt. 1 pt. 3 gi. were drawn out; how much oil remained in the vessel? Explanation. --3 gi. can not be taken from 2 gi. But the 1 pt. of the minuend Convenient Form equals 4 gi. and this added to 2 gi. gives 6 gi. 6 gi. — 3 gi. = 3 gi. 1 pt. from pt. 25 gal. 2 qt 1 pt. 2 gi. can not be taken. But 1 qt. is taken from 8 gal. 3 qt. 1 pt. 3 gi. 2qt., and changed to pints, giving 2 pt. Ans. 16 gal. 2 qt. 1 pt. 3 gi. Finally, 24 gal. — 8 gal. = 16 gal. Prove the work by adding the remainder to the subtrahend and comparing the result with the minuend. 5. A man built an average of 45 ft. 8 in. of fence a day for 16 da. ; how much fence did he build in 16 days? Convenient Form 45ft. Sin. 16 Explanation.— r- 8 in. X 16 = 128 in. = 10 ft. 8 in. 10 ft. 8 m. 45 ft. X 16 = 720 ft. 720 ^ns.730ft. Sin. 6. An iron rod, 4 yd. 2 ft. 8 in. long, was cut into 5 equal pieces ; how long was each piece? Convenient Form ^ . , ., « i.^ ^ .^ i..^ . ^ -^ Explanation. —4 yd. =12 ft. 12 ft. + 2 ft. 5 ) 4 yd. 2 ft. 8 m. = 14 ft. 14 ft. -*- 5 = 2, with a remainder of Ans. 2 ft. 114 in. ^ ^^' ^ f^- = ^^ i°- 48 in. + 8 in. = 56 in. §147. ExerciseB on Denominate Number Tables. standards of value 1. IIow many cents are there in 23 dimes? in $23.00? in 23 220 RATIONAL GRAMMAR SCHOOL ARITHMETIC 2. How many dollars are there in 248 cents? in 248 dimes? in 248 eagles? 3. How many half eagles are there in $245.00? in 240 dimes? 4. About what is the value of S50.00 in pounds sterling (Eng- lish money)? in Marks? in Francs? in Rubles? in Crowns? in Lira? 5. What are the more accurate equivalents of $100.00 in these same units? 6. I owe a debt of 32 M. 50 Pf . (pfennig = y^^ of a Mark). How much U. S. money will it take to pay this debt if I have to pay 24.5 cents for a Mark? 7. If I have to pay $4.94 for a pound sterling, how much U. S. money will it take to pay a debt in London of £65 8s.? WEIGHT 1. How many grains are there in (1) 12 dwt. ; (3) Si oz. ; (5) 9 oz. 12 dwt. (2) 3 oz. ; (4) 8 oz. 10 dwt. ; (6) 4 lb. 8 oz.? 2. A carat (of weight) usually means 32 troy grains. How many carats are there in a diamond weighing 13 dwt. 8 gr.? 3. How many carats weight are there in a diamond weighing 2 troy ounces? 4. The avoirdupois table is used for weighing iron, coal, and all coarser (less costly) articles. How many grains heavier is a pound of coal than a pound of silver? How many troy ounces ? 5. Express 8 lb. (av.) as ounces. Express 8 lb. (troy) as ounces. 6. How many ounces are there in 24 lb. of iron? in 48 lb. 6 oz. of iron ? 7. How many ounces are there in 3 cwt.? in 1 cwt. 40 lb.? in 5 cwt. 8 oz.? '8. Express as avoirdupois ounces the following avoirdupois weights : (1) 3 lb. 8 oz. ; (3) 86 lb. 15 oz. ; (5) 8 cwt. 13 lb. ; (2) 18 lb. 12 oz. ; (4) 1 cwt. 8 lb. ; (6) 9^ cwt. 9. Express as tons the following: (1) 40 cwt.; (3) 16000 1b. (5) 12500 1b.; (2) 30 cwt. ; (4) 9000 lb. (6) 13500 lb. COMPOUND DENOMINATE NUMBERS 221 10. Express the following as long tons : (1) 6720 lb. ; (3) 112 cwt. ; (5) 56 tons; (2) 19040 lb. ; (4) 952 cwt. ; (6) 39 cwt. 20 lb. 11. How many pounds are there in ^ of a ton of coal? in ^ of a long ton of pig iron? LINEAR MEASURE 1. How many inches make a rod? in 1 dhain? ^ 2. How many feet are there in 1 chain? 3. Express 3 rd. 2J yd. in inches. 2 rd. 1 yd. 2 ft. in inches. 4. Express 6408 inches in rods and yards. 5. A toy railroad track is 10 yd. long. How many feet of track are needed for it? 6. A 12-rod fence is made of 10-inch boards set vertically with no cracks between them. How many boards are needed? 7. The four sides of a garden are 10 rods, 1020 ft., 162^ ft., ■ and 1019 ft. 4 in. How many 10-inch boards will be needed to enclose it with a fence like that described in problem 6? 8. The tire of a bicycle is 88 inches long. How many times will the wheel turn in going 40 rods? 9. The tire of a wagon wheel is 15 ft. 9 in. long. How many turns will the wheel make in going 5 miles? SURFACE MEASURE 1. Express as square inches : (1) 3 sq. ft. ; (4) 2 sq. yd. 3 sq. ft. ; (7) 3^ sq. yd. ; (2) 3 sq. ft. 8 8q. in. ; (5) 6 sq. yd. 48 sq. in. ; (8) 45^ sq. ft. ; (3) 1 sq. yd. (0) 45 sq. ft. 108 sq. in. ; (9) 1 sq. rd. 2 Express: (1) 121 sq. yd. as sq. rd. ; (4) 80 s^. rd. as A. ; (2) 640 sq. rd. as A. ; (5) 1 sq. rd. as sq. ft. ; (3) 40 A. as sq. rd. ; (6) 1 A. as sq. yd. 3. Express: (1) 10 sq. ch. as A. ; (4) 1 Tp. as A. ; (2) 20 A. as sq. ch. ; (5) 1 A. as sq. rd. ; (3) 1 sq. mi. as sq. cb. ; (6) A. section as sq. rd. 4. How wide must a rectangle 80 rd. long be to contain 1 A.? 5 A.? 20 A.? 80 A.? ^22 flAttOHAL ORAMMAR SCHOOL ARttHMETlO 5. How wide must a rectangle 16 rd. long be to contain 1 A? How many sq. rds. more than 1 A. are there in a 14-rd. sqnare? G. How many square yards more than 1 A. are there in a 70- yd. square? 7. How many square yards less than 1 A. are there in a 69-yd. square? 8. How many square feet are there in 1 A.? 9. How many square yards more than 1 A. are there in a 21?- ft. square? CAPACITY 1. Express: (1) 2 qt. 1 pt. as pt. ; (4) 2 gal. as pt. ; (2) 1 qt. i pt. as gi. ; (5) 12 gal. as qt. ; (3) 3 qt. 1 pt. 2 gi. as gi. ; (6) 11 gal. 3 qt. as pt. 2. Express: (1) 24 pt. as qt. (4) G4 gi. as gal. (2) 48 gi. as qt. (5) 17 pt. as qt. and pt. (3) 76 pt. as gal. (6) 84 pt. as gal. and qt. 3. There are 231 cu. in. in a liquid gallon. How many cubic inches make a quart? a pint? 4. There are 268.8 cu. in. in a dry gallon. How many more cubic inches are there in a dry than in a liquid quart? 5. Express: (1) 3 bu. as pk. ; (4) 4 bu. as dry gallons; (2) 2i bu. as pk. ; (5) 18 dry gallons as bu. ; (3) 2i bu. as pk. ; (6) 20 pk. as pt. 6. What is the difference between the weight of a quart of barley in California and in Illinois? 7. How many bushels are there in a long ton (2240 lb.) of anthracite in Illinois? in a short ton? 8. My coal bin is 7'xl2'xl2'. How many bushels of anthracite will it hold? how many tons? (Count 1^ cu. ft. per bushel.) 9. About what part of a Winchester bushel is 1 cubic foot? 10. Express 3.5 bu. as pints; 160 pt. as bushels. 11. A grocer buys beans at $1.80 per bu. and sells them at 9^ a qt. Compare the buying and selling prices per bushel ; per quart. COMPOUND DENOMINATE NUMBERS 223 12. Find the error in the adage *'A pint's a pound'' if there are 7^ gal. in a cubic foot of water weighing 1000 ounces. TIME 1. Express: (1) 1 hr. as. sec. ; (4) 30 sec. as hr. ; (7) 1209600 sec. as wk. ; (2) 1 da. as sec. ; (5) 7200 sec. as hr. ; (8) 1000000 sec. as wk. da. and hr. (3) 1 wk. as sec; (6) 604800 sec. as da. ; (9) 1 wk. as minutes. 2. How many weeks in a common year? July 4 falls on Tues- day in 1905. On what day of the week will July 4, 1906, fall? Why? 3. How many months old are you this month? 4. How many days old are you to-day? How many hours in this many days? 5. A clerk earned $15 a week for 28 calendar months. How much did he earn in all? 6. A clerk saved $6 a week for 7 yr. How much did he save in all? 7. Illinois was admitted as a state in December, 1818. How many years and months old is the state of Illinois this month? • 8. How many years and months old is your native state. (See Table, page 39.) 9. On the Saturdays of April, 1905, the sun rose and set at Chicago ^t the times given here: DAY Date Sun Rosb Sun Set Saturday April 1 5:46 a. m. 6:23 p. m. Saturday April 8 5:34 a. m. 6:30 p. m. Saturday April 15 5:24 a. m. 6:37 p. m. Saturday April 22 5:14 a. m. 6:43 p. m. Saturday April 29 5:05 a. m. 6:50 p. m. Give the length in hours and minutes of each of these 5 days "from sun to sun," and also give in minutes the change per week of the days' length during April, 1905. 224 RATIONAL GRAMMAR SCHOOL ARITHMETIC 10. It is noon at 12 :00. How long in hours and minutes was the forenoon of April 1, 1905? the afternoon? How many min- utes longer was the afternoon than the forenoon? 11. Answer similar questions for Apr. 8th; 15th; 22d; 29th. 12. From the numbers of a common almanac compare the days' length of April with the same dates for October. Date Morning Twilight Evening Twilight 1905 Begins Ends Begins Ends Jan. 1st April 1st July 1st Oct. 1st 5:46 a.m. 4:10 a.m. 2:27 a.m. 4:26 a.ra. 7:25 a.m. 5:45 a.m. 4:32 a.m. 5:56 a.m. 4:42 p.m. 6:24 p.m. 7:35 p.m. 5:44 p.m. 6:21 p.m. 7:58 p.m. 9:40 p.m. 7:14 p.m. 13. From the table find how long morning twilight lasted on Jan. 1, 1905; on Apr. 1; July 1; Oct. 1. » 14. Answer similar questions for evening twilight. COUNTING THINGS 1. A stationer pays 25^ per dozen for lead pencils and sells them at 5^ apiece. What is his profit on a gross of pencils? 2. How many are "three score and ten"? How many years are "a score of centuries"? How many months? 3. Lincoln's Gettysburg address was spoken in November, 1863. The address begins "Four score and seven years ago," etc. To what date did Lincoln refer? 4. What is the value of 1 bale of paper at 12^ per quire? 5. What is the value of 3 great gross of steel pens at the rate of 2 pens for 5^? 6. What is the value of 10 great gross boxes matches at the rate of 2 boxes for 5^? §148. General Exercises. 1. Reduce 9 great gross to units. 2. How many great gross are there in 79,636 unitig? COMPOUND DEIJOAIINATE NUMBEUS 225 3. A dealer bought 3600 lead pencils at S2.00 per gross. He sold them at 25 cents per dozen. What was his profit? 4. Bought 2 gr. foot-rules at $.12 a doz. ; 7 gr. Spencerian pens No. 1 at $.07 a doz.; 8 gr. Eagle pencils No. 3 at $.30 a dozen. Find the amount of my bill. 5. A stationer bought 3 gr. boxes of paper, each box con- taining 6 reams. How many sheets did he buy? 6. A stationer sold 3 quires, 20 sheets of paper to one man, 18 quires to another, 4 reams, 15 quires to another. How many sheets did he sell in all? 7. There are 2 reams, 9 quires of paper in one package, 3 times as much in a second package, and 7 times as much in a third package as in the second package. How much paper is there in the second and third packages each? 8. In £28 there are how many shillings? how many pence? how many farthings? 9. In 18s. there are how many pence? how many farthings? 10. In £28 18s. 9d. 3 far., there are how many farthings? 11. In £342 15s. 6d. there are how many pence? 12. How many pounds, shillings, and pence are there in 28,643 pence? 13. How many feet are there in 78 yd.? how many inches? 14. How many inches are there in 78 yd. 2 ft. 6 inches? 15. How many yards, feet, and inches are there in 2838 inches? 16. How many ounces are there in 12 cwt.? in 10 pounds? 17. How many ounces are there in 12 cwt. 10 lb. 11 ounces? 18. How many hundredweight, pounds, and ounces in 19,360 ounces? There are two classes of problems to be solved in reducing denominate numbers to their equivalents in different units. In one class numbers expressed in larger units are to be expressed in smaller units. This is called reduction descending^ or reduction from Mgher to lotoer denominations. In the other class numbers expressed in smaller units are to be expressed in larger units. This is reduction ascending^ or reduction from lower to higher denominations. Problems 15 and 18 are examples of the second class, and prob- lems 13, 16, and 17 are examples of the first class. 226 BATIONAL GRAMMAR SCHOOL ARITHMETIC 19. Reduce 160 yd. 1 ft. 9 in. to inches. 20. Reduce 5781 in. to yards, feet, and inches. 21. Reduce 8 bu. 2 pk. 5 qt. to quarts. 22. Reduce 27 T. 18 cwt. 15 lb. to pounds. 23. Reduce 23 cu. yd. 14 cu. ft. to cubic feet. 24. Change £395 15s. sterling to dollars and cents. United States gold arid silver coins are .9 pure gold or silver and .1 copper. 25. What weight of pure silver is there in a silver dollar weighing 412^ grains? 26. The U. S. standard 5 dollar gold piece weighs 129 gr. What is the number of grains of pure gold in the standard five dollar gold piece? 27. The 5-cent piece weighs 73.16 gr. .75 of the weight of the coin is copper and .25 is nickel. What is the weight of copper in the 5-cent piece? of the nickel in the 5-cent piece? 28. The eagle weighs 258 gr. What is the weight of pure gold in the eagle? 29. How many standard gold dollars can be coined from 1 oz. of pure gold? 30. Express the following ratios : 1 franc : $.25; 1.25 : 1 franc; 1 mark : $.25; $.25 : 1 mark. 31. If .52 oz. of gold is worth 43s. 4d., how many ounces can be bought for £35 18s.? 32. A boy laid by a certain sum of money each week. At the end of 1 yr. 3 mo. 2 wk. he had saved 488.50. How much did he save each week? (Take 1 mo. = 4 wk. 2 days.) 33. A man changed $350, half into English and half into German money. How much of each kind of money did he have? 34. How many feet are 4 ch. 30 links? 35. What is the ratio of 1760 yd. to 1 mi. 32 rods? 36. The mast of a ship was 78 ft. 4 in. high. During a storm .3 of it was broken off. How high was the remaining piece? 37. A four-sided field had sides of the following lengths: 63 ch. 2 rd. ; 49 ch. 14 li. ; 53 ch. 1 rd. 16 li. and 38 ch. 24 li. How far is it around the field? COMPOUND DENOMINATE NUMBERS 227 38. A man walked | of the length of a breakwater, which was 1 mi. 243 rd. 5 yd. long. How far did he walk? 39. A knot, or geographic mile, equals 6086 ft. What is the speed in common or statute miles per hour of a vessel that runs 21 knots per hour? 40. A wheel, 12 ft. in circumference, makes how many revolu- tions in 1^ miles? 41. A telegraph wire is 14 mi. 140 rd. long, and is supported by 386 poles, which are placed at equal distances apart, n. pole being at each end of the wire. How many feet apart are the poles? 42. What is the cost of 18 A. 120 sq. rd. of land at $52 per acre? 43. A man owned 14 A. of land, and sold 1428 sq. rd. How many acres did he have left? 44. If 1000 shingles are needed to cover 100 sq. ft. of roof, how many shingles are required to cover a roof 40 ft. long and 25 ft. wide, at the same rate? 45. A lawn tennis court 120 ft. long and 85 ft. wide is to be surrounded by a strip of sod 15 ft. wide at each end and 8 ft. wide at each side. What will the sodding cost at $.35 per square yard? 46. Find the cost of a half section of land at $45 per acre. 47. How many square rods are there in a rectangular field 24 ch. 45 li. long by 16 ch. 34 li. wide? 48. How many cubic inches are there in a tank containing 120 gal. ? how many cubic feet? 49. How many cubic feet are there in a block of stone 5 ft. x 4 ft. X 6 inches? 50. How much must I pay for a board 16 ft. long and 8 in. wide, the board being 1 in. thick and lumber costing $35 per thousand? (A board foot means 144 sq. in. of surface, not over 1 in. thick.) 51. I bought 3 boards 12' long, 6" wide, 8 boards 16'long,and 9" wide, and 2 boards 10' long, 12" wide and all were 2" thick; what did the whole cost at $30 per thousand? 52. What is the value of a straight pile of wood 16 ft. long, 8 ft. wide and 6 ft. high, at $7.50 per cord? 228 RATlOKAL GRAMliAE SCHOOL AHltHMETIC 63. What is the weight of a pile of oak boards 14 ft. long 8 ft. 4 in. wide and 5 ft. high, at an average weight of 54 lb. per cubic foot? 54. At $.75 a load of 1 cu. yd. what will be the cost of remov- ing a pile of earth 60 ft. long 24 li. wide and 1 yd. high? 55. In one year the quarries of Minnesota yielded 4,000,000 cu. ft. of sandstone. What was this worth, at $.76. a perch? 56. Find the cost of building a stone wall 150' x 10' x 2^, at $3.50 a perch? 57. A bin 12' x 8'6" x 5' is filled with wheat. What is the weight of the wheat at 60 lb. per bushel? 58. What is the value of a straight pile of pine slabs 32'x 7'x 4', at $3.25 per cord? 59. A grocer paid $4.50 for a barrel of vinegar (31| gal.), and sold it at 5^ per quart. What was his profit? 60. A jug contained 214 cu. in. of molasses. How much did it lack of containing 1^ gallons? 61. What will 1 mi. of right of way for a railroad cost at $00 per acre, the width of the right of way being 100 feet? 62. What will be the cost per mile for railroad ties at 1.45 apiece, the ties being laid one every 2 ft. (Omit the odd tie.) 63. Find the cost of the rails for 1 mi. of the road, the rails weighing 77 lb. per linear yard and costing $35 per long ton. 64. Find the cost of fencing 1 mi. on both sides of the track, placing posts costing $.25 apiece 16 ft. from center to center, and using wire weighing 2160 lb. per mile, and 30 lb. of staples @ 4f The fence is to be 4 wires high, and labor costs 21^ per rod of fence. (Here count the odd post.) 65. Reduce .865 gal. to smaller units. Solution.— .865 x 4 qt. = 3.46 qt. ; .46 X 2 pt. = .92 pt. ; .92 X 4 gi. = 3.68 gills. Am. 3 qt. 3.68 gills. . 66. Reduce .168 gal. to smaller units. 67. If oil is $.11 per gallon, what will be the cost of tnree 42- gallon barrels of kerosene? 68. If a certain spring regularly yields 25 gal. daily, how many barrels of the capacity of 31^ gal. would it fill in 20 days? COMPOUND DENOMINATE NUMBERS 229 69. A merchant bought 16 gal. 3 qt. of syrup at 38^ a qt., and sold 27 qt. for $12, 18 qt. for $9, and the remainder at 36^ per quart. Did he gain or lose, and how much? 70. Beduce 2 pk. 7 qt. 1 pt. to the decimal of a bushel. Solution.— 1 pt.= .5 qt. 7.6 qt.= ^ pk. = .9375 pk. 2.9375 pk = o 2.9375 . noAory^ u u 1 — 3 — bu. = .734375 bushels. 4 71. Reduce 3 pk. 5 qt. 1 pt. to bushels. 72. How many bushels are there in a bin 14 ft. long 7 ft. 6 in. wide and 5 ft. 8 in. high? 73. A farmer sold 6 loads of corn, each load averaging 36 bu. 2 pk. at 35 cents per bushel. What did he receive for the whole? 74. A boy had a bushel of hickory nuts, and sold 3 pk. 7 qt. 1 pt. What fraction of a bushel had he left? 75. If 9 bu. of potatoes cost $4.80, what is the average cost per peck? 76. A bin contained 5376.25 cu. in. of rye. How much did it lack of containing 3^ bushels? 77. How many oz. of quinine will be required to prepare 12 gross of 3-grain capsules? 78. A coal dealer buys two car-loads of coal each weighing 67,200 lb., at $4.50 a long ton, and sells it at $5.75 a short ton. What is his gain? THE METRIC SYSTEM §149. Historical. The metric system is a decimal system of weights and meas- ures adopted by the French Government soon after the French Revolution of 1789. The aim of the system is to base all measures upon an invariable standard, and to secure the simplest possible relations between the different units of the system. The unit of length, which is fundamental to the whole system, is called the 7neter. It was attempted to make the meter 1 ten- millionth of the length of the part of a meridian of the earth, which reaches from the equator to the pole, called a quadrant of the earth's meridian. The meridian of the earth was measured, and TTnTTArTTnr of the quadrant was obtained. A platinum bar equal to this length was very accurately cut and stored in the Government archives as the official standard of reference. 230 RATIONAL GRAMMAR SCHOOL ARITHMETIC Later measurements of the earth's meridian showed the former length of the meridian to be incorrect. The length of the meter as obtained from the erroneous measures was, however, retained, and the length of this bar is the standard meter. From it all the other units of the system are derived. The metric system is used for all purposes in France, and for nearly all scientific purposes in Germany, England, and the United States. The length of the meter is 39.37079 in., or about 1.1 yards. §150. Tables of Metric Measures. Fig. 136 shows a scale graduated along the upper edge to 16th8 of an inch, and along the lower edge to lOOOths of a meter, called millimeters. ' i 'l ' l ' I' I'I ' I'I'I ' I ' l 'I'I 'I 'I' I ' M I ' l '| 'I'| M '|' M | ' I '|M'| ' ''[ V a' 3' 4' ,i.ij.iii.i.i.i,iii.u,i.i?i,iii.i.i:i.i,i,i.i!i,i,i.i.i?i.i,i,i.iii.i.i,i?i,i.i.i.iii.i.i,fl Figure 136 Decimal parts of the standard units are denoted by the Latin prefixes, abbreviated in each case to a single, small letter: milli, meaning 1000th written m. centi, meaning 100th written c. deci, meaning 10th written d. Multiples of the standard units are denoted by the Greek prefixes, abbreviated in each case to a single, capital letter: deka, meaning 10 times written D hekto, meaning 100 times written H. kilo, meaning 1000 times written K. myria, meaning 10000 times written M. Table of Measures of Length 10 millimeters (mm.) = 1 centimeter (cm.) = about .4 in. 10 centimeters = 1 decimeter (dm.) = about 4.0 in. 10 decimeters = 1 meter (m.) = about 1.1 yd. 10 meters = 1 dekameter (Dm.) = about 32.8 ft. 10 dekameters = 1 hektometer (Hm.) = about 328 ft. 10 hektometers = 1 kilometer (Km.) = about .62 mi 10 kilometers = l myriameter (Mm.) =s f^l^ut 6,21 mi COMPOUND DENOMINATE NUMBERS Table of Surface Measure 231 100 sq. millimeters (mm*.) 100 aq. centimeters 100 sq. decimeters 100 sq. meters 100 sq. dekameters 100 sq. hektometers r= 1 sq. centimeter (cm*.) = 1 sq. decimeter (dm^. ) = 1 SQ. METER (m^.) = 1 sq. dekaraeter (Dm^.) ^ 1 sq. hektometer (Hm*.) = 1 sq. kilometer (Km^. ) SQ.INCH SQ. CENTI- METER FIGUKE 137 The cm*. = .155 sq. in. The m«. = 10.764 sq. ft. The Km*. = 247.114 A. 1 m* =1 centare (ca.) Table of Land Measure 100 centares = 1 Are (pronounded air) (a.) 100 ares = 1 hektare (Ha.) Table of Measures of Volume The standard unit of volume is the cubic meter = 35.314 cu. ft. = about 1.2 cubic yards. 1000 cubic millimeters (mm'.) = 1 cu. centimeter (cm'.) 1000 cubic centimeters = 1 cu. decimeter (dm'.) 1000 cubic decimeters = 1 cu. meter (m'. ) etc., etc. Table of Measures of Capacity The standard unit of capacity is the liter (leeter). It is equal to 1 cu. decimeter, and equivalent to .908 dry quarts. 10 milliliters (ml.) := 1 centiliter (cl.) 10 centiliters 10 deciliters 10 liters 10 dekaliters 10 hektoliters = 1 deciliter (dl. ) = 1 liter (1.) = 1 dekaliter (Dl.) = 1 hektoliter (HI.) = 1 kiloliter (Kl.) Table of Measures of Weight The standard unit of weight is the gram, which is the weight of 1 cu. centimeter of distilled water at its temperature of greatest density (39.1° F.). 232 RATIOJ^AL GRAMMAR SCHOOL ARITHMETIC 10 milligrams (mg. ) : 10 oeotigrams 10 decigrams 10 grams 10 dekagrams 10 hektograms 10 kilograms 10 myriagrams 10 quintals : 1 centigram (eg.) : 1 decigram (dg.) : 1 gram (g.) : 1 dekagram (Dg.) : 1 hektogram (Hg.) : 1 kilogram (Kg.) : 1 myriagram (Mg). : 1 quintal (Q.) : 1 metric ton (T.) 1 centigram = .15432 grain 1 gram = 15.432 grains 1 kilogram = 2.20462 lb. 1 metric ton = 2204.621 lb. §151. Metric and TJ. S. Equivalents. The equivalents will be of assistance in changing the metric to the common system of measures. 1 m. = 39.37 in. lKm.= .6214 mi. 1 mile = 1.6093 Km. 1 yard = .9144 m. 1 m2. = 1.196 sq. yd. IKm^. = .3861 sq. mi. lare = .0247 A. 1 square yard = .8361 m^. 1 square mile = 2.59 Km^. 1 acre = 40.47 a. 1 m«. = 1.308 cu. yd. 1 stere = .2759 cord 1 liter IHl. -i 1.0567 liquid qt .9081 dry qt. ^ j 2.8376 bu. i 26.417 liq. gal. 1 g. = 15.432 grains 1 Kg. = 2.2046 lb. avoir. 1 metric ton = 1.1023 T. 1 cubic yard 1 cord = .7645 m8. = 3.624 St. 61022CU. in. = 1 1. 1 liquid qt. = .9436 1. Idryqt. =1.1011. 1 bushel = .3524 HL 1 grain = .0648 g. 1 lb. avoir. = .4536 Kg. 1 T. = .9072 T. PROBLEMS WITH THE METRIC USTITS 1. Take a smooth lath, or plane one, lay a straight strip of paper beside the lower edge of the metric rule, Fig. 13G, page 230, and with a sharp pencil transfer the graduations from the rule to the strip of paper, and then from the strip of paper to the lath. COMPOUND DENOMINATE NUMBERS 233 Continue the centimeter graduation marks along your lath until you get a meter stick (= 100 centimeters). How does this meter stick compare in length with a yard stick? 2. Explain which digit in 6.8752 m. stands for m. ; which for dm. ; which for cm., and which for millimeters. 3. Find by measurement the length and the width of your schoolroom in meters, and write the result of your measurements in meters and decimals of a meter. 4. How many centimeters long, wide, and thick is your arith- metic? 5. Express the length of your pencil in centimeters. 6. A sidewalk is 2112 m. long, how many kilometers long is it? 7. What is the cost of 158 cm. of ribbon at $.45 a meter? 8. How many steel rods, each 16 cm. long, can be cut from a rod 7.68 m. long? 9. From a piece of wire, 98.52 m. long, a piece .047 Km. was cut. How long was the remainder? 10. A bicycle track is .807 Km. long. How many meters would one go in riding around the track 5 times? 11. Express 1 Km*, in square dekameters; in square meters. 12. Find the number of square meters in the surface of your schoolroom. Express the same in square dekameters. 13. Express the area of your desk in square decimeters; in square inches. How do the two compare? 14. Using the decimeter as a measure, find the number of square centimeters in the surface of your geography. 15. Express the same in square decimeters; in square inches. 16. Find the area of the door in square meters. What part of an are is this area? 17. A certain plot of ground contains 20 m*. How many square feet in the plot? 18. With the aid of your meter stick, lay off a square meter upon the floor. Within this area, lay off a square yard. Note results. 19. With the aid of your decimeter measure, dra\«^ a square decimeter. In the corner, draw a square centimeter. How many square centimeters could be drawn within the square decimeter? 234 RATIOKAL GRAMMAR SCHOOL ARITHMETIC 20. What is the area of a floor 8.4 m. long and 5 m. wide? 21. A lot is 7.64 m. wide and .033 Km. deep. What is the area? 22. What is the value of a piece of land 8 Dm. long and 6. 5 Dm. wide, at $32 an «re? 23. How many ares in a street 1.5 Dm. wide and 2.48 Km. long? 24. How many square meters in a floor 700 cm. long and 500 cm. wide? 25. A man had 3 pieces of land as follows: 8.4 Ha.; 3846 m'., and 2.5 Km^. How many hektares of land are there in all? 26. 20 liters equal how many centiliters? what part of a Dl.? of a Kl.? Find the difference between 12 1. and 12 quarts. 27. Using your decimeter measure as a basis, model a cubic decimeter or liter. 28. Using your square centimeter as a basis, model a cubic centimeter. What is the relation of a cubic centimeter to a gram? How many cubic centimeters in a liter? 29. How many cubic meters are there in the volume of your schoolroom? 30. What is the difference between an ordinary ton and a metric ton (tonneau)? 31. A gram is the weight of 1 cm^. of distilled water. What is the weight in grams of 1 1. of distilled water? 32. A liter of water weighs nearly 2^ lb. What is the weight of 6 1. of water in pounds? in grams? 33. A package of silver weighs 2.58 grams. What is its weight in grains? 34. How many pounds in 1 Q.? in 6 quintals? 35. How many 2 gr. capsules will 5 g. of quinine fill? 36. What would be the cost of the quinine at 10^ per dozen capsules? 37. What is the cost of 5500 Kg. of coal at $6.50 per ton? 38. What is the cost of 2 Q. of sugar at $.08^ a kilogram? 39. If a book weighs 3.2 Dg., how many such books will weigh 1.792 kilograms? PERCENTAGE AND INTEREST 235 40. A box contains 2500 packages of quinine, each package weighing 1.25 g. How many hektograms does the contents of the box weigh? 41. A vesse}, 18 cm. long, 14 cm. wide, and 22 cm. high, is filled with lead, which weighs 11.35 times as much as water. What is the weight of the lead? (1 cu. ft. of water weighs 62.5 lb.) PEBCEVTAGE AND IVTEBEST §152. Percentage. ORAL WORK In a city block there are 100 houses. 50 of them are stores; 40 are dwelling houses; 5 are warehouses; 3 are hotels, and 2 are banks. 10 of them are built of brick; 30 are stone, and 60 are frame houses; 35 of them are 3 stories high, 25 are two stories, and the rest are one story high. 1. What part of the houses are stores? How many hundredths of the houses are stores? Note. — "How many per cent?" "What per cent?" means "How many hundredths?" 2. How many per cent of the houses in the block are stores? 3. i equals how many per cent? 4. What part of the houses are 2 stories high? What per cent of them are 2 stories high? 5. What per cent are warehouses? hotels? banks? dwellings? 6. What per cent are brick? stone? frame? 7. What per cent are 3-story buildings? 1-story buildings? 8. What part of the houses are warehouses? ^V equals how many per cent? 9. What part of the houses are dwellings? f equals how many per cent? 10. What part of the houses are frame houses? | equals how many per cent? A sidewalk tOO rd. long is made up of 10 rd. of stone walk, 25 rd. of cement walk, 40 rd. gravel walk, 50 rd. board walk, and 75 rd. cinder walk. 11. What per cent of the entire walk is stone? What per cent cement? gravel? board? cinder? 236 RATIONAL GRAMMAR SCHOOL .ARITHMETIC 12. What per cent of the whole was the stone and the cement? the cement and the cinder? the gravel and the board? the stone and the board? In a certain nursery there are 100 oak, 75 ash, 50 maple, 30 elm, 25 hickory, 15 poplar, and 5 walnut trees. 13. What per cent of the trees were oak? What part were oak? 14. What per cent of the trees were ash? What part were ash? 15. What per cent of the trees were maple? elm? hickory? poplar? walnut? 16. $2 equals what part of $8? 9 bu. equals what part of 12 bu.? 8 hr. equals what part of 24 hr. $6 of $9? 17. To how many hundredths of a number are the following fractional parts of that number equal: i? i? I? I? V f? F F I? t\^ tV tV irV? A? 2V 18. Express the following as hundredths : 3. 9. T.fl.1.6. 1.5.1.3.6. 3 T(r> ¥7> lfJ59 tlTj €i 6> \J9 Tf> t» «> 8> T15- Definitions. — The words per cent mean hundredth, or hundredths. The sign, " %," is a short way of writing percent, or hundredths; thus, 2%, 8%, 33J%, mean xJo. 185* fii. and are read, "2 per cent," "8 per cent," "33} per cent" The number (as 2, 8, 33} ), written before the sign, " %," is called the rate per cent. It is well to recall that we have the foUoi^fing ways of writing such numbers as 6 per cent: (1) 6 per cent; (3) 6% ; (3) 6 hundredths; (4) jSq ; and (5) .06. The sign, " %," is merely an abbreviation for ^J^, or .01. 19. Of 100 kinds of animals of a neighborhood, 27 die, 18 migrate and 10 hibernate as winter comes on. What per cent die? migrate? hibernate? 20. 10 kinds store food, 20 remain and feed abroad and 10 kinds appear only in winter. What per cent store food? feed abroad? appear only in winter. WRITTEN WORK Review §80, pp. 123-125, as oral work. Note. — Whenever you can answer a problem orally, do so. Form the habit of using your pencil only when it is necessary. 1. In a schoolroom containing 40 pupils, | of the pupils were girls. How many girls were there? PERCENTAGE AND INTEREST 237 2. In the last problem, what per cent of the pupils were girls? 3. In the month of September, 12 da. were cloudy. What per cent of the days were cloudy? 4. Express the equivalents of the following fractions as per cents, or hundredths : i; I; 14;*; I;*; iV; A; A; if- 5. In a sample of soil weighing 37 oz., 18 oz. were sand. What per cent of the soil was sand? 6. A sample stick of timber, weighing 19 lb., contained 2 lb. of water. What per cent of the weight of the timber was due to the water it contained? Solution.— (1) Annex zeros after the numerator and divide thus: 19 )2.00 But .IItV = lliV%- Why? 7. What per cent of a yard is 1 ft.? 9 in.? 15 in.? 28 in.? 35 in.? 1 inch? 8. The first number of these pairs is what per cent of the second: (1) 8 of 150?' (4) ^ of 60? (7) 32.91 of 263.28? (2) 13 of 700? (5) 27i of 600? (8) i of i? (3) 7i of 12? (6) 38.45 of 769? (9) | of 12J? 9. Change the following per cents to their fractional equiva- lents (fractions in their lowest terms) : 5%; 8%; 8^%; 12^%; 12%; 16%; 16|; 22^%; 67i%; 87^%. 10. Change to their decimal equivalents the following: 7% ; 6i% ; 83i% ; 87^% ; 1*% ; 2i% ; 3 % ; yV of 1% ; jh of 1%. 11. At the beginning of a school year, the lung capacity of a boy was 161 cu. in. By the middle of the year it was «7% larger. How many cubic inches had his lung capacity increased? 12. The lung capacity of a boy at the beginning of the year was 160 cu. in., and at the middle of the year it was 166.4 cu. in. What was the per cent of increase? 238 RATIONAL GRAMMAR SCHOOL ARITHMETIC 13. The standing of the several clubs in the National League during one season was determined from this table : Club Games Won Games Lost Per Cent Woh Pittsburg 90 49 64.7% Philadelphia.... 83 57 Brooklyn 79 57 St. Louis 76 64 Boston 69 69 Chicago 53 86 New York 52 85 Cincinnati 52 87 Find for each club what per cent of the total number of games played were won. Garry results to the first decimal place, as shown for Kttsburg. 14. What per cent of the farm, Fig. 4, p. 7, is the cornfield? the wheatfleld? the meadow? the south oat field? the pasture? the lot occupied by the house and grounds? 15. 60% of the value of a mill is $5400; what is the value of the mill? 16. 3% of a school of 1200 children were absent; how many children were absent? 17. I pay 12% of the value of the property I occupy as rent every year. My rent is $240 a year; what is the value of the house? 18. A house was damaged by fire, and an insurance company paid the owner $840 damages, which was 40% of the original cost of the house ; what was the original cost? §153. Algebra. ORAL WORK 1. 50 equals what per cent of 100? of 200? of 150? of 500? 2. 2 equals what per cent of 4? of 8? of 6? of 20? 3. Any number equals what per cent of a number twice as large? 4 times as large? 3 times as large? 10 times as large? 4. X equals what per cent of 2.r? of 42;? of 3a;? of 10a;? 5. 7a equals what per cent of 14a? of 28a? of 21a? of 70fl? 6. 13 equals what per cent of 17? of 28? of 45? of 55? 13 13 X 100 A^?A 76^8 SOLUTION.- -j^ = ^^^^^ = ^ = _X^ = 76ft %. PERCENTAGE AND INTEREST 239 7. 13a; equals what per cent of 17a;? of 28a;? of 4:5a;? of 55a;? 8. X equals what per cent of y? of z? of m? of ^? Note— — =-^ = 100—%. 1^ 100 1^ ^ WRITTEN WORK 1. What is 2% of $200? of $375? How is 2% of any number of dollars found? 2. How is 2% of $a found? What is 2% of la? of $J? of $x? of $6x? 3. How is 6% of any number found? What is 5% of the number a? of b? of x? of z? of 82;? 4. How is 12J% of any number found? What is 12^% of a? of X? of 10a;? 5. IIow is any per cent of a number found? G. How can you express a as hundredths? x, as hundredths? 9a;? 12y? 4:5z? 7. Express these numbers as hundredths: 16; 29; a; x; m; 16a;. 8. What is r% of 160? of 350? of a? of ^? of C? of 27W? Definitions. — The result of finding a given per cent of any amount, or number, is called the percentage. The amount, or number, on which the percentage is computed is often called the base. 9. Calling the percentage, jh the rate per cent, r, and the base, 5, show by an equation how to find jy, from b and r. 10. Replacing the symbols in your equation by the words for which they stand, translate the equation into the common lan- guage of percentage. This translation, properly made, is the fundamental principle of percentage. Principle. — The percentage equals the product of the base and rate divided by 100, or more briefly, (I) i'=r^,- All of percentage is contained in this equation, called a formula, because it formulates {expresses in brief form) a law. Multiplying both sides of the formula by 100, we have {A) 100i?-ir. 240 RATIONAL GRAMMAR SCHOOL ARITHMETtC Now divide both sides of this equation by r, and write the second number (see §69, p. 97) first, and obtain (note that (II) b = lME. 11. Translate (II) into words. How would you find the base (b) if the percentage (p) and rate (r) were given? 12. Divide both sides of (A) by b, write the second number first, and make a rale for finding r when p and d are given. §154. Oain and Loss. 1. A merchant paid $16 for a suit of clothes. At what price must he sell the suit to gain 25 per cent? 2. A stationer pays $1.80 a dozen for blank books, and retails them at a profit of 66f %. At what price per book does he sell them? 3. A grocer buys eggs at wholesale at 10^ a doz.., and retails them at a profit of 30%. Supposing there is no loss due to break- age or spoiling, at what price per dozen does he sell them? 4. Allen paid $1 for a sled, and sold it at a loss of 20%. How much did he receive for the sled? 5. From a box containing 200 oranges, 80% were sold in one day. How many oranges were sold? 6. What is the percentage on $640 at 20%? at 28%?? at 35%? at 6i%? at 12i%? at 87^%? 7. A farm, costing $4400, was sold at a gain of 18% ; for how much did the farm sell? 8. A farm sold for $5192, which was 18% more than was paid for it. How much was paid for it? Solution.— The 18% here means 18% of what was paid for the farm (the cost price). $5192 is then equal to what per cent of the cost? The statement may be written thus: 1.18 times the cost price = $5102, or, more briefly, thus: 1. 18a? = 85192. Hence, .r =—^^70. Divide and find the value of x. 9. A center fielder threw a ball 90 ft., which was 12% farther than the third baseman threw it. How far did the third baseman throw it? Suggestion. --First answer the question, 12% of what? PERCENTAGE AND INTEREST 241 10. A boy bought oranges at the rate of 5 for 3jJ, and sold them at the rate of 3 for 5^ ; what was his rate per cent of gain or loss? 11. A merchant in shipping 150 crates of eggs lost 25 crates by freezing. What per cent did he lose? 12. Hats, costing $2.75, sold for $3.50. What was the per cent of profit? 13. An automobile, costing $750, sold for $1250. What was the per cent of profit? 14. In a certain battle, 32,000 men were engaged, and 35% of all engaged lost their lives. How many lost their lives? 15. A horse, costing $88, sold at a loss of 40%. For how much did the horse sell? (40% of what?) 16. I paid $35 for a bicycle, and sold it the next season for 60% less than I paid for it. What was the selling price? 17. A cow gave 13 lb. of milk Nov. 1, and 15.6 lb. on Nov. 15. What was the rate % of gain in 14 days, in her daily yield of milk? 18. Make similar problems on the table, page 16. 19. The increase in population in one year in a certain town was 2000, which was 6J% of the population at the end of the year? What was this population? 20. The death rate the same year was 1.8% of this popu- lation. How many deaths occurred? 21. A man lost 27% of the books of his library by fire. He lost 540 books. How many books did his library contain before the fire? 22. Anthracite coal cost $8.50 a ton in Nov., and rose 15% in 1 mo. What was the price of anthracite coal after the rise in price? 23. I sold a lot for $600, at a loss of 20%. What did the lot cost? 24. Find the cost of a piano which sold for $187.50 at a loss of 25%. 25. I bought a tennis outfit for $35, and sold it at a loss of 15%. Find the selling price. 26. Carpet, marked at $1.25, sold at a reduction (discount) of 6f %. Find the selling price. 242 RATIONAL GRAMMAR SCHOOL ARITHMETIC 27. A man lost 20% of his money and after losing 10% of the remainder had $3,600 left. How much had he at first? 28. A dealer sells a cask of 30 gal. of oil, and receives $30. In delivering it 4.5 gal. were spilled. What per cent was spilled, and how much should the dealer pay back? 29. Goods damaged by fire were sold at a loss of 40%. The amount received was $555. What was the original cost? 30. A suit of clothes was marked at $45, which was 50% moi-e than the cost. It sold at a reduction of 20% from the marked price. What was the per cent of profit on the first cost of the suit? §155. Meteorology. The number of clear, cloudy, partly cloudy, and rainy or snowy days for the first 4 mo. of 1903, at Chicago, are here tabulated : 1903 CLEAR CLOUDY g^^J^ ^nSw^ Jan. 9 15 7 9 Feb 10 11 7 9 Mar 12 14 5 11 Apr 10 13 7 13 1. What per cent of the number of days of Jan. were clear? cloudy? partly cloudy? rainy or snowy? 2. Answer similar questions for Feb. ; for March; for April. 3. What per cent of the total number of clear days of 1903 to May 1st fell in Jan.? in Feb.? in March? in April? 4. Answer similar questions for cloudy, partly cloudy and rainy, or snowy days. 5. The highest velocities (speeds)in miles per hour of the wind in Chicago for each of the 12 mo. (beginning with Jan.) of a certain year were: 50; 48; 57; 70; 54; 45; 56; 47; 52; 58; 52; 58. What per cent of the highest velocity for April was the highest velocity for Jan.? for Feb.? for each of the remaining months? 6. The total movement, in miles, of the wind in Chicago for each of the 12 mo. of a certain year was: in Jan. 12,736; 10,279; 13,999; 12,820; 12,356; 10,900; 11,231; 9,839; 11,834; 13,148; 13,583; 15,476. What per cent of the total wind movement in Dec. was the total movement for each of the other months? PERCENTAGE AND IKTEREST 243 The table below gives the height in feet (elevation), above sea level and the total wind movement in miles for an entire year, of a number of important weather signal stations. Block Island is on the Ehode Island coast, and Mount Tamalpais'is a mountain station near San Francisco. Roseburg (Oregon) is the quietest place, as to wind movement, reported in the United States: Station Elevation Wind Movement Mount Tamalpais 2,375 163,203 Block Island 26 152,838 Chicago 823 145,193 Cleveland 762 128,566 New York , 314 127,267 Buffalo 767 125,042 Boston 125 98,755 Philadelphia 117 95,319 St. Louis 567 84,482 New Orleans 51 74,299 Louisville 525 70,396 Washington 112 63,629 Roseburg (Oregon) 518 30,471 7. What per cent is the wind movement of Chicago of that of each of the other placed in the list? 8. What per cent is the elevation of Chicago of that of each of the other places? 9. 100 green oak leaves weighed 100 grams. When thor- oughly dried they weighed 39.5 grams. The dried leaves were then burned, and the ash weighed 2.4 grams. What per cent of the leaves was dry solid? mineral matter (ash)? 10. 100 green elm leaves weighed 60 grams. 38^%, by weight, was dry solid, and 2f % was mineral matter. What was the weight of the dry solid? of the mineral matter? 11. A school garden, 15' x 40', was seeded as follows: 25%, potatoes; 16|% of the remainder, peas; 20% of the remainder, beans; 25% of the remainder, lettuce; 33^% of the remainder, radishes; 20% of the remainder, parsley, and the rest in beets. Hot many square feet were seeded to beets? 244 BATIOKAL ORAHMAB SCHOOL ARITHMETIC 12. The precipitation (rainfall) at Chicago, in inches, for Jan., Feb., Mar., and Apr. for 1903 was 1.09, 3.03, 1.67, and 3.77, and the averages for these same months for 33 yr. are 2.05, 2.11, 2.52, and 2.73. Find the rate per cent of excess, or defi- ciency, for each of the four months of 1903, on the 33-yr. average. 13. Problems may be- made on the following table of data relating to Chicago weather: 1902 Temperature Precipitation Weather Monthly Mean Mean for 31 Years In Inches for 3S Yr. Clear Days Pair Days Caoudy Days Jan. . . Feb. .. Mar. . . Apr. . . May .. June . . July . . Aug. . . Sept. . . Oct. . . . Nov. . . Dec. .. 25.2 20.8 38.6 46.4 59.0 64.2 72.4 68.4 60.8 55.2 47.0 26.5 23.8 25.9 34.3 46.4 56.5 66.6 72.3 71.1 64.4 53.1 38.5 29.0 .66 1.53 416 2.26 5.08 6.45 5.78 1.44 4.83 1.45 2.03 1.90 2.08 1 2.30 2.56 2.70 8.59 3.79 3.61 2.83 2.91 2.63 2.66 1 2.71 10 12 6 10 10 7 12 15 9 14 8 7 14 9 13 12 19 16 13 10 8 10 13 8 7 7 12 8 2 7 6 6 13 7 9 16 §156. The Almanac. Note. — In this section "length of the day" means the length of day- light, or the time interval from sunrise to sunset. 1. On Jan. 1, at Chicago, the sun rose at 7 hr. 29 min., and set at 4 hr. 38 min., and on July 1, it ;rose at 4 hr. 28 min., and set at 7 hr. 39 min. The length of the day on Jan. 1 equals what per cent of the length on July 1? 2. On Jan. 1, the length of the day in St. Louis was 103.8%, and in St. Paul 96.4%, of the length of the same day in Chicago. Find the day's length in each of these cities. 3. On Julyl, in the same places, the day's lengths were, respect- ively, 97.9% and 102.2%, of its length in Chicago. Find the day's length in each of these cities on this date. 4. Make and solve similar problems for the calendar pages of a common almanac. PERCENTAGE AND INTEREST 245 5. On the first of Feb., of March, of April, of May, of June, of July, of Aug., of Sept., of Oct., of Nov., and of Dec, the day's lengths in Chicago were 108%, 122.3%, 137.8%, 152*6%, 163.2%, 163.5%, 157.9%, 144.6%, 129.2%, 132.5%, and 102.6% of the day's length on Jan. 1. Find the day's length on the dates named. 6. On vertical lines, one for each'month, plot these rates to scale and draw through the points a smooth freehand curve. 7. Similar problems may be obtained from the calendar pages of a CQmmon almanac. 8. From the almanac find what per cent the period from *'moon rises" to "moon sets" on the first of some month is of the same period on the 15th or 2dth of the same month. 9. Find what per cent the time from full moon to new moon is of the time from new moon to first quarter; from new moon to second quarter, or full moon. §157. Geography. Find these per cents to one place of decimals. 1. Eefer to the table of p. 30, and find what per cent of the population (1900) of your state was in elementary and secondary (high) schools. 2. Compare the result of problem 1 with the corresponding results for any other states. Data for the following problems will be found on pp. 30 and 39. 3. What per cent of the area of the United States (without outlying territory) is the area of the North Atlantic division? of the Western division? 4. What per cent of the total population of the United States (continental) is the population of the North Atlantic division? of the Western division? 5. Find what per cent the area and population of your state is of the area and population of the division to which your state belongs. 6. Find the per cent of increase of population of your state from 1890 to 1900. 246 RATIONAL GRAMMAR SCHOOL ARITHMETIC 7. Find the per cent of increase of population of the (conti- nental) United States during the same time. 8. Find the per cent of increase in population from 1890 to 1900 of the division to which your state belongs. 9. The following table shows the growth in territory of the United States and the dates when the additions were made. Find the per cent of increase of each addition on the date of acquiring to 1899 inclusive. Acquisition i Datb '^a Ml* Original territory 1783 827,844 Louisiana purchase 1803 1, 182. 752 Florida /. . . 1819 59.268 Texas 1845 371 ,063 Mexican purchase 1848 522,568 Texas purchase 1850 96,707 Oadsden purchase 1853 45,535 Alaska 1867 590,884 Hawaii 1898 6,449 Porto Rico ^ 3,600 Philippine Islands j^ 1899 114,000 Guam J 200 Isle of Pines 1899 882 10. The area in millions of square miles of the surface of the earth is 148.18; of this surface 108.77 is water, and 39.41 land. Find what per cent of the surface of the earth is water; land. The land surface equals what per cent of the water surface? The following table contains the actual lengths of the coast lines of the continent and also the lengths that would be needed to enclose them if they wero solid, with smooth outlines, also the ratio of low land to high land : CONTINENT LE^THS North America 24,040 South America 13,600 Asia 30,800 Africa 14.080 Europe 17,200 Australia 7,600 Length, Ratio Low to IF Solid Highland 10,380 6i:9 9,030 9J:2J 13,780 10^:13 11,760 61:141 6.630 4|:1J 5,860 3}a pehcentage akd ikterest 247 11. Find what per cent of each number in column 2 equals the corresponding number in column 3. What does each per cent mean? Which continent has the longest coast to defend in com- parison with its area? 12. Express the given ratios of column 4 of low land to high land in per cent, and tell what the per cents mean? The table below contains the heights, in feet, of mountains and peaks of the world : Mt. Everest 29,002 Aconcagua 23,910 Chimborazo (volcano) 20,500 Kilimanjaro Mts 20,000 Mt. Logan 19,500 Karakoram Mts 18,500 Orizaba (volcano) .... 18,312 Mt. St. Ellas 18.100 Popocatapetl(volc.). . 17,784 Mt. Wrangell 17,500 Mt. Blanc 15,744 Mt. Shasta 14,350 Longs Peak 14.271 Pikes Peak 14,147 Mt. Etna (volcano) .. 10,875 Rocky Mts 10,000 13. The height of Pikes Peak equals what per cent of that of Mt. Everest? of Mt. Logan? of Chimborazo? of Mt. Shasta? 14. Answer other similar questions on the table. 15. What per cent of North America (16,130,269 sq. mi.) is drained through the Mississippi basin (1,250,000 sq. mi.)? through the St. Lawrence basin (360,000 sq. mi.)? the Columbia basin (290,000 sq. mi.)? the Colorado basin (230,000 sq. mi.)? 16. What per cent of the length of the Mississippi River (4,200 mi.) is the length of the St. Lawrence (2,000 mi.)? of the Columbia River (1,400 mi.)? of the Colorado River (2,000 mi.)f Yukon River (2,000 mi.)? The following table gives the lengths, in miles, and the areas of basins, in square miles, of 24 of the longest rivers of the world: AREA OF RivEB Length basin Mississippi 4,200 1,250,000 Nile 3.900 1,300,000 Amazon 8,600 2,500,000 Yangtzekiang . . 3.300 650,000 Obi 8,000 1,000,000 Yenisei 3.000 1,400,000 Congo 3,000 1,500,000 Niger 2,900 1,000,000 Hoangho 2,800 890,000 Amur 2,700 780,000 Lena 2,700 900.000 La Plata 2,500 1,250,000 Arba of Rtveb Length qasin Mackenzie 2,400 680,000 Volga 2,400 590,000 St. Lawrence . . 2,000 360,000 Yukon 2,000 440,000 Brahmaputra . . 2,000 426,000 Colorado 2,000 230,000 Indus 2,000 325,000 Euphrates 2,000 490,000 Danube 1,900 320,000 Rio Grande 1,800 225,000 Ganges 1.800 450,000 Orinoco 1,500 400,000 248 RATIONAL GRAMMAR SCHOOL ARITHMETIC 17. What per cent of the length of the Mississippi equals that of the Nile? What per cent of the area of the Mississippi basin equals the Nile basin? 18. Solve other similar problems on the table. The following table contains the areas, in sq. mi., the alti- tudes, in ft. (above sea level), and the greatest depths, in ft., of the great lakes of the world : Lakb Area Altitudb depth Superior 31,200 602 1,008 Huron 23.800 581 700 Michigan 22,450 581 875 Erie 9.960 573 212 Ontario 7,242 248 738 Victoria 22,167 4,000 620 Winnipeg 9,400 710 72 19. The area of Lake Michigan equals what per cent of that of Lake Superior? 20. Similarly compare the altitudes and greatest depths of these lakes. 21. Solve similar problems on the table. §158. Commission. Definition. — Commission is a sum of money paid by a person or firm called the principal, to an agent for doing business for the principal It is usually reckoned as some per cent of the amount of money received or expended for the principal. 1. An agent sells 250 bu. @ 80^- on a 2% commission. How much money was due the agent? 2. An agent bought for his firm 1000 bbl. of apples @ *1.25, and received a commission of 5%. How much money did he receive? 3. What is the commission on 100 baskets of peaches @ 50^, the rate of commission being 3%? 4. What is the commission on the following sales: (1) 80 A. land @ $75, rate of commission being 5%? (2) City property sold @ $8500, rate of commission 4^%? (3) 40 bu. tomatoes @ 75^, rate of commission 10%? (4) 1000 bu. wheat @ 95^, rate of commission 1^%? PERCENTAGE AND INTEREST 249 (5) 100 doz. eggs @ 25 jJ, rate of commission 12%? (6) 1000 bu. green vegetables @ 90^, rate of commission 12^% ? (7) $500 collected debts, rate of commission 15%? (8) $1200 bad debts collected, rate of commission 20%? 5. Find the commission on $1850 at 2^% ; at 3|% ;. at 12^%. 6. A principal sent his agent $2652 to be used for the firm after deducting (subtracting) the agent's commission of 2%. How much money was used for the firm? 7. A clothing house sent its agent $1224, which included a sum of money to be invested and 2% commission on this sum for the agent. What sum was to be invested? 8. 5% commission on a certain ainount of money was $684.20. What was the amount? (Statement: .05a; = $684.20. Find a;.) Definition. — A shipment of goods sent to an agent to be sold is called a consignment. 9. A consignment of 4560 bu. of wheat was sold by an agent at 78f^ per bushel. What was the agent's commission at 1| per cent? 10. A commission agent sold the following consignment of goods: 20 doz. eggs at 14i^; 40 lb. creamery butter at 21^; 36 lb. cheese at 13:^^; 80 lb. chickens at 12^^; 8 doz. live chickens at $3.75; 4 bbl. apples at $3.25; 16 boxes oranges at $2.25; 4 boxes oranges at $2.50; 12 boxes grape fruit at $2.75; 25 bunches bananas at $1.25; 12 boxes lemons at $2.75; 16 bbl. potatoes at $4.25. Find the agent's commission at 6^ per cent. 11. An agent's commission at 2|% on a certain collection amounted to $97.16. What was the amount of the collection? §159. Trade Discount. Definition. — A discount is a certain rate per cent of reduction (sum thrown off) from the prices at which articles are listed or quoted. The discount is usually allowed for cash payments or for payment within a specified time. 1. A retail merchant buys silk at $1.20 a yard. He is allowed a discount of 10 ^fc for cash. He pays cash. How much does the silk cost him? Solution.— $1.20— 10% of $1.20 = $1.08. 250 RATIONAL GEAMMAR SCHOOL ARITHMETIC 2. A publisher sells 60 books at $1.50 and allows 20% dis- count. How much does he receive for the books? Merchants and manufacturers often publish expensive catalogues containing their price lists of articles, whose prices change rap- idly. When prices fall, instead of publishing new lists, they mark off an additional discount. For example, the price of a certain article may be catalogued thus: $60, discount, 20%, 10%, 5%. This means a 20% reduction, then a 10% reduction on the reduced price, and then a 5% reduction on the second reduced price. It , would be computed thus : $60 $48 $43.20 .20 .10 .05 $12.00 $4.80 $2.1600 $60 -$12 = $48. $48 -$4.80 = $43.20. $43.20 -$2.16 =$41.04. Note.— Notice this is not the same as a discount of 20% + 1^?^ + 5^ (= 35% ). Wherein is it different? 3. A New York merchant sells to a customer goods marked thus: Price, $3500; disc't, 10% 60 da. and 5% for cash. What must the customer pay to settle the bill by cash payment? Note. — The customer gets the benefit of both discounts. What would the customer have paid if he had been given a single discount of 15 per cent? 4. A bill of plumber's supplies was marked thus: Price: $7.50, disc't. 20% and 7% and 5%. How much did the supplies really cost? 5. Compute the amount of money needed to settle the follow- ing bills if paid in cash or within the shortest time mentioned: (1) $45; discount 25% 60 da. and 10% 5 days. (2) $180; discount 16|% 30 da., and 10% 10 da., and 5% cash. (3) $1800; discount 30% and 10%, and 7% cash. (4) $54; discount 12^%, and 8%, and 5% 30 days. (5) 46 T. coal at $5.75; discount 10% and 2% cash. (6) 50 men's suits at $18; discount 20% and 5% 10 days. (7) 4 gross tablets at 50^ per doz. ; discount 20% and 7% cash. 6. To what single rate of discount is a discount of 20% and 6% equivalent? Suggestion.— Take a base of $10Q. PERCENTAGE AND INTEREST 251 §160. Marking Goods. In marking his goods a merchant nses the key word "harmo- nizes." Rewrites the seUinff price shove a horizontal line, and the cost price helow it J using the letters h-a-r-m-o-yi-i-z-e-s in order for the digits 1-2-3-4-5-6-7-8-9-0. For example a book sells at J>2.50 and costs $1.75. The mark would be t-t- . hio 1. Using the key "harmonizes," interpret the following cost marks and find the per cent of profit for each cost mark : (1) ^; (2) i^, (3) ^-^, (4) Vm, (5) ^. ^ ' ma ^ ^ ' or ^ ^ ^ hio ^ ' rss ^ ' rrn 2. Complete these marks so that the selling price may be 40% greater than the cost price : (5)f^; (6)^^^; (7)^^. ^ ' hao ^ ' rao ^ ' mos 3. Articles bearing the following selling marks are marked to sell at a profit of 33^% ; fill in the cost mark, using the same key as above : (1) ^; m 4^; (3) ^; (4) ^; (5) '^', (6) ^^ 4. Using the same key, mark articles costing the following prices to sell at a profit of 37^% : (1) 40^; (2) 72jf; (3) $1.68; (4) $2.88; (5) $6.40; (6) $8.24. 5. Supply complete cost marks for articles sold at 50% profit the selling prices of which are as follows: (1) 90^; (2) $1.50; (3) $2.50; (4) $3.66; (5) $7.50; (6) $8.10o 6. Solve problems 4 and 5, using the key *'black horse." 7. Choose a key word and with it solve problems 4 and 5. 8. A merchant wishes to mark his goods so that he may drop 10% below the marked price and still make 20% of the cost price. Using the key "harmonizes," how must he mark articles costing the following prices : (1) 50^? (2) 60^? (3) $1.20? (4) $2.50? (5) $6.40? 252 RATIONAL GRAMMAR SCHOOL ARITHMETIC §161. Interest. oral work Review §81, pp. 125-26. The method of §81 is known as the six per cent method. In reckoning interest the year is regarded as containing 12 mo. of 30 da. each. 1. A man is charged $0 for the use of $100 for 1 yr. What per cent of the sum borrowed ($100) equals the sum ($6) he is charged for its use? 2. A man is charged $21 for the use of $350 for 1 yr. The sum charged equals what per cent of the sum borrowed? Definitions. — Interest is money charged for the use of money. It is reckoned at a certain rate per cent of the sum borrowed for each year it is borrowed. When money earns 3, 6, 7, or 10 cents on the dollar annually (each year) the rate is said to be3%,6%,7%,orl0% per annum (by the year) and the rate per cent is said to be 3, 6, 7, or 10. 3. At 6% per annum, how much interest does $360 earn in 1 yr.? in 3 yr.? in 4^ yr.? in 2| yr.? in t years? 4. Make a rule for computing the interest on any sum of money at 6% when the time is in years. 5. At 6% per annum, how much interest does $1 earn in 1 yr.? in 2 mo.? in 1 mo.? in G da.? m 12 da.? in 18 days? 6. When the time is in months, how may the interest on any sum of money at 6% per annum be computed? 7. How may the interest at 6% per annum on any sum of money be computed when the time is given in months and days? in years, months, and days? Definitions. — ^The sum of money on which the interest is computed is called the principal. The principal plus the interest is called the amount. Since the borrower must not only pay the interest on the bor- rowed principal, but also return the principal, the debt he must discharge (pay) is the amx>unt. 8. How long will it require any principal (say $1) to amount to twice its value ($2) or to double itself at 6% per annum? 9. Give the reasons for these statements : Any principal at 6%— (1) doubles itself in 200 months; (2) earns j^^ of itself in 2 mo. or 60 days. PERCENTAGE A^D IKTEEESt 253 10. Let / stand for the interest on $p at r % for t yr. and let i denote the interest on %p at 6% for t yr. Explain the meaning of the equations : WRITTEN WORK To find the Interest. 1. What is .07 of $450? What is the interest on $450 at 7% for 1 yr.? for 2 yr.? for 5 yr.? for 3| yr.? for 4J yr.? for t years? 2. What is .08 of $1250? What is the interest at 8% on $1240 for 1 yr.? for 3 yr.? for 5}J yr.? for x years? . 3. $640 was on interest at 5% from July 1, 189G, to July 1, 1900. Find the interest. 4. $1800 was on interest at 7% for 3 yr. 7 mo. 21 da. Find the interest. Convenient FoBtis I. Interest computed first at the given rate. $1800 principal .07 rate int. for 1 yr. whole years int. for 3 yr. int. for 6 mo. int. for 1 mo. int. for 15 da. int. for 6 da. ^58.85 int. for 3 yr. 7 mo. 21 da. 11. Interest computed first at 6%. .18 = int. on 11 for 3 yr. at 6% .035 = int. on |1 for 7 mo. at 6% .0035 = int. on |1 for 21 da. at 6% .2185 = int. on %\ for 3 yr. 7 mo. 21 da. at 6% 1800 $126.00 3 $378.00 } of $126.00 63.00 J of }of 63.00 10.50 10.50 5.25 J of 10.50 2.10 1748000 2185 393.3000 = int. on $1800 for given time at 6% 65.55 = int. on $1800 for given time at 1% $458.85 = int. on $1800 for given time at 1% SUOOESTIONS FOR II.— (1) If the interest on $1 for 1 yr. is 6^, what is the interest for 3 years? (2) If the interest on $1 for 2 mo. is 1^, what is the interest for 7 mo.? (3) If the interest on $1 for 6 da. is 1 mill, what is the interest for 21 da.? 254 RATIOKAL aftAjtfMAft SCflOOL A<ttMETlC 5. How much interest must I pay for the use of $600 for 1 yr. 5 mo. 24 da. at 7 per cent? 6. Find the amount of $300 for 2 yr. 4 mo. 25 da. at 4^ per cent. 7. Find the interest and the amount under the following con- ditions : (1) 1700 for 1 yr. 7 mo. 15 da. at 3 per cent. (2) $400 for 2 yr. 9 mo. 27 da. at 3 per cent. (3) $210 for 2 yr. 5 mo. 28 da. at 6 per cent. (4) $150 for 1 yr. 11 mo. 13 da. at 7 per cent. (5) $280 for 1 yr. 6 mo. 19 da. at 4^ per cent. (6) $3G0 for 1 yr. 4 mo. 5 da. at 4 per cent. (7) $260 for 2 yr. 3 mo. 11 da. at 3^ per cent. (8) $500 for 1 yr. 11 mo. 14 da. at 7 per cent, (9) $300 for 2 yr. 7 mo. 12 da. at 5 per cent. (10) $625 for 3 yr. 9 mo. 18 da. at G per cent. To find the Principal, 8. What principal at 8% will furnish $16 interest in. 2 years? Suggestion.— What interest will $1 produce at 8% in 2 yr.? How many dollars will yield 816 at 8% in 2 years? 9. What principal at 8% will produce $30 interest in ^ years? 10. What principal at 6% will amount to $112 in 1 yr. G months? Suggestion.— -What is the amount of $1 at C% interest for 1 yr. 6 mo.? 11. Find the principal which will yield $61.25 interest in 3 yr. 6 mo. at 7 per cent. 12. Find the principal which will amount to $972.40 in 3 yr. 2 mo. 12 da. at 4^ per cent. 13. Make a rule for finding the principal when the rate, time, and interest are givren. 14. Make a rule for finding the principal when the rate, time, and amount are given. PERCENTAGE AND INTEREST 255 15 r Supply the correct value for the letter in each of the fol- lowing cases : Rate Time Interest Principal Amount (1) 6 % 1 yr. 7 mo. 15 da. $19.50 P A (2) 8 % 3 yr. 3 mo. 18 da. $169.02 P A (3) 8i% 4 yr. 7 mo. 21 da. / P $998.50 (4) 7 % 6 yr. 8 mo. 16 da. / P $198.42 (5) 4 % 2 yr. 6 mo. 18 da. $122.50 P A (6) 5 % 1 yr. 10 da. $176.00 P A (7) 4i% 90 da. $32.50 P A To find the Rate. 16. At what rate per cent will $320 yield $34 interest in 2 yr. 9 months? Suggestion.— How much interest will $320 yield in 2 yr. 9 mo. at 1%V At what rate per cent then will the same sum yield $34 interest in 2 yr. 9 months? 17. At what rate will $780 yield $486.60 in 5 yr. 8 months? 18. A man invested $2000 for 2 yr. 7 mo. 27 da. and received $2638 at the end of this time. What rate per cent of interest did his investment earn for him? 19. A man bought 120 A. of land at $85 and sold it 2 yr. 8 mo. later for $100 per A., after having received $900 in rents from it and having twice paid taxes on it at 75 cents per acre. What was his annual rate per cent of profit? 20. Make a rule for finding the rate when the principal, time, and interest are known. 21. Find the correct value to the first decimal place for the letter in each of the following problems: (1) $580.00 1 yr. 5 mo. $46.50 r% (2) $1280.00 3 yr. 10 mo. $500.00 r% (3) $798.45 2 yr. 8 mo. 15 da. $258.65 r% (4) $3698.50 1 yr. 5 mo. 19 da. $568.75 r% To find the Time. 22. How long will it take $80 to earn $14 interest at 4% annually? Suggestion. — How much interest will $80 earn in one year at 4% ? In how many years then will $80 earn $14 at the same rate? 256 RATIONAL OBAMMAR SCHOOL ARITHMETIC 23. How long will it take $125 to earn $57.50 interest at 8^ per annum? 24. At 7%, how long will it take $648 to yield $69.84? Note.— When the time results in decimals of a year the decimal may be reduced to months and days by the method of problem 65, p. 228. 25. How long will it take $750 to yield $750 interest at 8%? 26. How long will it take $10 to double itself at 6%? at 7%? 27. How long will it take $975 to amount to $1225 at 5%? Suggestion.— What is the total interest? What is the interest on 1975 at 5^ for 1 year? 28. Make a rule for finding the time, T^ required for a given principal, P, to amount to a given sum, J, at a given rate per cent, r? 29. Make a rule for finding how long it will take a given principal to earn a given interest at a given rate per cent. 30. Find the time, T!, in years, months and days under the conditions stated in each of the following problems : PmNciPAii Rate Interest Time (1) $66.00 7 % $28.60 T (2) $460.00 6 % $31.05 T (3) $750.00 Hlo $147,475 T (4) $1260.00 5 % $213.15 T (5) $2460.00 4 % $321.44 T 31. Supply the value for which each letter stands in the proh- lems of the following table : Principal Rate Time Interest Amount (1) WOO 6% 3 yr. 3 mo. 8 da. I A (2) $175.00 h% 4 yr. 9 mo. 15 da. I A (3) , $800.00 7% T I $926.00 (4) $475.00 8% T $142.50 A (5) $1266.00 4% T I $1349.85 (6) P 6% 5 yr. 7 mo. 27 da. $509.25 A (7) $1575 30 v% 1 yr. 4 mo. 18 da. I $1662.461 (8) $728.25 ^fo T $209,736 A W $864.75 rfc 2 yr. 1 mo. 15 da. $69.29 A PEROEKTAGE AKD INTEREST 32. Complete, to mills, the following interest table : Interest Table: Principal «100. 257 Time Rate 3% 4% 5% 6% 7% 1 da $.008 $.011 §.014 $.017 $.019 2 da 3 da 4 da 5 da 6 da 1 mo 1 2 mo 1 3 mo 6 mo 1 vr 1 I 33. Compute by the table the interest on $758 at 7% for 4 mo. 12 da. ; for 7 mo. 8 da. ; for 1 yr. 3 mo. 10 days. §162. Algebra. 1. Compute the interest on $750 at 5% for each of the following times: (1) 1 yr. ; (2) 2 yr. ; (3) G^ yr. ; (4) 31 yr. ; (5)25|yr.; (G) a; yr. ; (7) / years. 2. Find the interest and the amount on $1250 for each of the following : (1) 5%, 1 yr. (2) G%,2iyr. (3) 7%, 6 yr. (4) 6%, 3iyr. (5) 4%,7fyr. (0) 3%, f yr. (7) r%, ^yr. (8) r%,a:yr. (9) r%, nyr. 3. Denote the interest on a certain principal, P, by 7, the rate by r, and the time (in years) by /, write an equation showing liow to find I from P, r, and /, and translate into words the meaning of the equations. 25B RATlOKAL 6IIAMMAR SCttOOL AlllTHMEtlC Solution.— (I) (1) I=Px^Xt, or ^^) ^= 100 - ^' Translated into words: (1) means, ^'Interest equals the product of the principal, the rate divided by 100, and the time (in years)." (2) and (8) mean, ''Interest equals the product of principal, rate, and time, divided by 100." 4. Calling A the amount, / the interest, and F the principal, write an equation showing how to find A from / and F, 5. Write an equation to show how to find F from /, r, and /, and state in words the meaning of the equation. Prt If we multiply both sides of the equation, 1 = -jrrr, by 100 we have the equation, 100 J = Prt. Now to show how to find P from J, r, and t, divide both sides by rt, and write the second member on the left. We then have: (II) P=M_^. G. State in words the meaning of formula (II). 7. Show, by proper multiplications and divisions of equation (I) (3), that the rate r may be found from P, /, and ty by the equation, (III) r = -^^. 8. State in words the meaning of formula (III). 9. Show from (I) (3) that the time t may be found from /, P, and r by the formula, (IV) t^l^ 10. State in words the meaning of formula (IV). 11. Solve by formulas I-IV the following problems: (1) F = $64, r = 8, and jf = 2^; find 7. (2) /= $24, r = 6, and if = H; find P. (3) / = $75, F = $850, and ^ = 2 ; find r. (4) /= $120, F = $G00, and r = 10; find t. (5) 7= $230,131, F = $722, and t = ^; find r. PERCEKTAQE AND INTEREST 269 §163. Promissory Notes. Definitions. — A promissory note is a written promise, made by one person or party, called the maker, to pay another person or party, called the payee, a specified sum of money at a stated time. The snm of money for which the note is drawn is called tJieface value, or the face, of the note. The date on which the note falls due is called the date of maturity, and the time to run is the time yet to elapse before the note falls due. /c?5^ Kankakee, III., ^ne 3, '/903, Jrme/u t/auajBlier date I promise to pay to the order of S^, ^^M. ySadef^, ^wo S^^unt/?^ •::^V^ ^^ 00 100 Dollars, for value received, with interest at ofx per cent per annum from date. 1. lVh« is the maker of the above note? the payee? What is the face of the note? the date? the rate of interest? the date of maturity? the time to ru^? 2. A promissory note, unless otherwise specified in the note, draws interest on its face value at the rate mentioned in the note from the date of the note until it is paid. Compute the interest and the amount on the foregoing note if it was paid Sept. 1, 1903. 3. Find the interest on the following note, paid Aug. 30,1903: $875. Urbana, III, May 18, 1897. One year after date I promise to pay to James Black, or order, Eight Hundred Seventy-five and ^ Dollars, at Busey's Bank, for value received, with interest at the rate of seven per cent per annum from date. Due May 18, 1898. HENRY OSBORN. Note. — To find the time for which interest is to be computed, pro- ceed thus: Convenient Form Explanation. — Date of payment, 1903 2 10 2 mo. 10 da. = 1 mo. 40 da. Date of note. 1897 5 18 1 «io. 40 da. — 18 da. = 1 mo. 22 da. ' 1903 1 mo. = 1902 13 mo. Time, 5 8 22 1902 18 mo. — 5 mo. = 1902 8 mo. yr. mo. da. 1902 — 1897 = 6 yr. 2G0 RATIONAL GRAMMAR SCHOOL ARITHMETIC 4. Find the amount of each of the following notes : Face Rate Date or Note Date or Payment Interest Am'nt (1) $250 6% Mar. 12, 1899 Jan. 1,1902 (2) $635 7% Nov. 20, 1896 July 15, 1900 (3) $2400 5i% Dec. 12, 1899 Jan. 8, 1903 (4) $3865 4i% Oct. 13, 1901 Sept. 7, 1903 (5) $3640 Si% Aug. 17,1900 Mar, 3,1903 §164. Discounting Notes. Definition. — Discount is a deduction from the amount due on a note at the date of maturity. In some cases promissory notes do not draw interest. The fol- lowing is an example: John C. Cannon purchased a self-binding harvester from A. R. Crow for $120 and gave him^the following note in payment: 1120. Peoria, III, June 20, 1899. Eighteen months after date, for value received, I promise to pay to A. R. Crow, or order. One Hundred Twenty and ^ Dollars, without interest until due. Due Dec. 20, 1900. JOHN C. CANNON. 1. On S0pt. 20, 1899, Crow sold the note to Adams at such a price that Adams received his purchase money and 8% interest on it until the date of maturity (Dec. 20, 1900). How much did Adams pay for the note? Suggestions.— Any principal at 8% will amount to 110% of itself in 1 yr. 3 mo. Why? Hence, 1120 = 110% of what number? Or, better, 1.10a? = 1120. Find the value of x. Definitions.— The sum of money which, at the specified rate and in the time the note is to run before falling due, will amount to the value of the note, when due, is called the present worth of the note. The difference between the value of the note, when due, and the present worth is called the true discount. The bank discount of a note is the interest upon the value of the note when due, from the date of discount until the date of maturity. 2. If Crow had sold the above note to a banker at a discount of 8%, the banker would have computed the interest at 8% on $120 from the date of sale (Sept. 20, 1899) until the date of maturity. How much would he have received for the note? Find the difference between the true and the bank discount of the note. PERCENTAGE AND INTEREST 261 3. C. A. Thomas bought a road wagon of J. K. Duncan, giving' the following note in payment: 165. Pekin, III, Sept. 10, 1900. Two years after date I promise to pay J. K. Duncan, or order, Sixty -five Dollars, value received, with interest at 7% per annum. C. A. THOMAS. Due Sept. 10, 1902. On March 10, 1901, Duncan sold this note to a bank at 7^% discount. How much did Duncan receive for the note? Note. — Remember, bank discount is computed on the amount of the note when due, for the time to run from date of sale. 4. Counting money worth 7%, how much did Duncan receive for the wagon? 5. A man bought a horse, giving in payment his note for 1 yr. for 185, dated Feb. 2G, 1903, and drawing interest at 7% from date. Two months later 'the holder of the note discounted it at a bank at 6%. What was the discount? What did the bank pay for the note? 6. Find the bank discount and proceeds on the following notes : FACE RATE DATE OP Note M^iTi^PX ^t'^A^?' oV^ Dis^ Maturity Sale Disc, count (1) $60 6 % Apr. 6. 1897 Oct. 6, 1898 June 15, 1897 7 % (2) $275 6i % Aug. 7, 1899 Nov. 7, 1901 Mar. 13, 1900 6 % (3) $350 7J% Sept. 10, 190U Dec. 10. 1902 Jan. 15, 1901 6% (4) $700 8 % Feb. 15, 1901 Nov. 15, 1903 June 20, 1901 6 % (5) $858 6 % Jan. 1. 1900 July 15, 1903 May 19, 1900 7 % (6) $1260 5 % Feb. 28,1896 Aug. 31, 1900 Aug. 8,1896 6 % ..... (7) $1800 5i% June 19, 1897 Aug. 19, 1901 Dec. 29, 1898 6% (8) $2450 7% Nov. .18, 1899 Feb. 28, 1902 Dec. 1, 1900 ^5^% (9) $3865 6 % Dec. 20, 1901 Nov. 20, 1902 Feb. 12, 1902 5 % (10) $8600 4J% Oct. 18,1900 Jan. 18, 1904 Apr. 2,1901 i% §165. Partial Payments. Definition. — When a note or bond is paid in part the fact is acknowledged by the holder by his writing the date of payment, the sum paid, and his signature on the back of the note or bond. This is called an indorsement Partial payments are made only (1) on notes which read, **0n or before, etc.," (2) by private agreement between the maker and the holder of the note. 262 RATIONAL GRAMMAR SCHOOL ARITHMETIC For calculating the balance due on a note or bond on which partial payments have been made, nearly all the states have adopted the following rule, known as '*The United States Kule of Partial Payments," which has been made the legal rule by a decision of the Supreme Court of the United States: Rule. — Fmd the amount of the principal to the time when the payment or the sum of the payments equals or exceeds the interest due; subtract from this amount the payment or the sum of the payments. Treat the remainder as a new principal and proceed as before. ILLUSTRATIVE EXAMPLES 1. Find the balance due on the following note at maturity: $1250. 'Chicago, III. May 21, 1900. On or before two years after date I promise to pay to the Order of P. A. Hopper Twelve Hundred Fifty and ^^ Dollars at the Corn Exchange National Bank, for value received, with interest at 6 per cent per annum. Due May 21, 1902. JOHN P. MILLER. The indorsements on the back of this note were as follows: Nov. 21, 1900 $80.00 Feb. 21, 1901 $10.00 May 21, 1901 $150.00 Feb. 21, 1902 $500.00 PERCENTAGE AKD INTEREST 263 Solution by Rule Principal on May 21, 1900 11250.00 Interest for 6 mo. on $1 .03 Interest due Nov. 21, 1900 $ 37.50 Amount Nov. 21, 1900 (date of first payment of $80) §1287.50 First payment 80.00 New principal Nov. 21, 1900 11207.50 Interest for 3 mo. on $1 .015 Interest to Feb. 21, 1901 (date of second payment of |10) .... $ 18.11 Payment being less than interest no settlement is made. 11207.50 Interest for 6 mo. on $1 .03 Interest to May 21, 1901 (date of third payment of $150) $ 36 23 Amount May 21. 1901 $1243.73 Sum of second and third payments ($10 + $150) 160.00 New principal May 21, 1901 $1083.73 Interest for 9 mo. on $1 .045 Interest to Feb. 21, 1902 (date of fourth payment of $500) . . . $ 48.77 Amount Feb. 21, 1902 $1132.50 Fourth payment 500.00 New principal Feb. 21, 1902 $ 632.50 Interest for 3 mo. on $1 .015 Interest to date of maturity, May 21, 1902 $ 9.49 Balance due at maturity, May 21, 1902 $ 641.99 2. The following payments were made on a $650 note, bearing 7% interest and dated April 20, 1901: July 30, 1901 $75.00 • Jan. 15, 1902 $15.00 Aug. 12, 1902.. ..$175.00 Jan. 1, 1903 $50.00 Find the amount due April 20, 1903.. 3. A note of $2800, dated Feb. 23, 1900, and bearing 7% interest, carried the following indorsements: Feb. 23, 1901 $100.00 July 16, 1901 $50.00 • Jan. 1, 1902 $800.00 July 15, 1902 $85.00 Nov. 28. 1902 $380.00 Find the amount due Feb. 23, 1903. APPLICATIONS TO TKAK8P0KTATI0N PKOBLEMS bo ^ 13 ^ OOT*Cf 009*09 00T*9C o rd ^ ^ o S3 bo c o QQ 5=1 <»1 e ^ o 0) 0) o ^ o •^ tl rid P bO w •^ o ^ -^ O 0) o 264 APPLICATIONS TO TRANSPORTATION PROBLEMS 265 3. Give the following distances in feet : (1) Between the centers of the leaders; (2) Between the centers of the rear leader and of the front driver; (3) Between the centers of the drivers; (4) Between the centers of the rear driver and of the trailer; (5) Between the centers of the trailer and of the front wheel of the tender (tank) ; (6) Between the centers of the front two wheels of the tender; (7) Between the centers of the front leader and of the trailer; (8) Between the centers of the front leader and of the trailer ; (9) Between the centers of the front leader and of the rear tender truck wheel. 4. In the drawing the distance between the centers of the engine truck wheels is \y\ The actual distance between the cen- ters of the engine truck wheels is 85". . Find the scale of the drawing of Fig. 138. 5. Using the scale found in problem 4, find by measuring with a ruler graduated to IGths of an inch the actual distance from the center of the front engine truck wheel to the front tip of the pilot (cow-catcher). 6. The actual distance from the center of the rear tender truck to the rear end of the tender is 5' 3^". What is the actual dis- tance from the front tip of the pilot to the rear end of the tender? 7. In the drawing it is If" from the top surface of the rails to the top of the smokestack. How high is the top of the smoke- stack above the rails? 8. In the drawing, the distance from the top surface of the rails to the top of the boiler is 1". How high is the top of the boiler above the top of the rails? 9. How high is the top of the engine cab above the top of the rails? The top line of the tender? The top of the lantern con- taining the headlight? 10. Find the size of the windows of the engine cab. 11. Give the distance in inches (1) between the nearest points 266 RATIONAL GBAMMAIi SCHOOL ARITHMETIC on the rims of the drivers ; (2) between the nearest points on the rims of the leaders; (3) between the nearest points on the rims of the front tender wheels. 12. IIow long are the radii of the leaders shown in the cut? How long are the circumferences of these wheels? 13. Compute the radii and the circumferences of the tender ■ wheels. 14. Find the circumference of the drivers; of the trailers. 15. When the drivers tiirn round 240 times a minute, how fast does the engine go? 16. How many times do the leaders turn while the drivers turn round once? 17. The weights written beneath indicate the number of pounds of the weight of the engine which is borne by the different pairs of wheels. Find the total weight of the engine. 18. The weight of the- empty tender is 43,000 lb. When the tender is loaded it carries 10 T. coal and 7000 gal. of water. A cubic foot of water weighs 62^ lb., and contains 7^ gal. Find the total weight of the loaded tender. 19. The engine shown in the cut drew a train of 3 sleepers, averaging 93,000 lb. ; 5 passenger coaches, averaging 78,000 lb. ; an express car, weighing 34,000 lb. ; and a mail car, weighing 75,000 lb. What was the total weight of the train, including both engine and tender? 20. The force (pull) exerted by an engine to draw a train is different for different speeds. For a speed of 10 mi. an hour it has been found that on straight, level track an engine must exert a force of 4f lb. for each ton of weight of the train, including the weight of both engine and tender. Find the force required to draw the train of problem 19 under these conditions. 21. For a speed of 15 mi. an hour a force of 5^ lb. per ton of train weight is needed. What force will draw the* train of prob- lem 19 at this speed? 22. Following are the forces for different speeds from 20 to 75 mi. an hour. Find the pulling (tractive) force to be exerted by APPLICATIONS TO TRANSPORTATION PROBLEMS 2G7 the engine to draw the train of problem 19 at each indicated speed : Speed Force in Lb. PERT. Tractive Force Speed Force in Lb. PER T. Tractive Force 20 6i 50 lU 25 .n 55 12; 30 8 60 13 85 B| 65 131 40 9 70 14i 45 lOi 75 15J 23. Find the difference between each number and the number next above it in the table. What do you find? V 24. In the equation F= -^ + 3, let F stand for the force in pounds per ton and let V stand for the speed in miles per hour. Let F = 20 in the equation. Find F by dividing 20 by 6 and adding 3 to the quotient. Compare your result with the number of column 2, and in line with 20. What do you find? 25. Let V = 25. Find F and compare with the number of the table m line with 25. 26. Let V equal other numbers of columns 1 or 4 and find F for each speed. Definitions. — Replacing V in this way by numbers like 20, 25, and so on is called substituting for V the numbers 20, 25, and so on. Performing the operations indicated in the equation and obtaining the nmnber for F is called find- ing the value of F. 27. How does the equa- tion say that the force in lb. per ton {F) needed to draw the train can be obtained when the speed (F) of the train is known? 28. The speeds of the table (problem 15) are plotted to scale on the horizontal line of Fig. 139, and the forces, in lb. per ton of load, are plotted to a different softle on the vertical parallels. The points X to 12 are 8 5^ 4?' 20 25 30 35 'Mo 45 50 55 eio 65 70 7S 80 85 M Speed (V) in mi. per hr. Scale /fl"i Smi.per hr Figure 139 268 RATIONAL GRAMMAR SCHOOL ARITHMETIC the upper ends of the vortical lines, whose lengths represent the successiye numbers of columns 2 and 5 of the table. Place the edge of a ruler along these points, or stretch a thread taut just over them. How do the points seem to lie? 29. Notice the horizontal and the vertical scales and make a drawing like that of Pig. 139 to a scale 4 times as large. Y 30. Assuming that the same law, /'= — + 3, relating force and speed, holds also for a speed of 80 mi. per hour, substitute F= 80 in the equation and compute F, the force in pounds per tonneeded to draw the train 80 mi. per hour. Make a similar computation for F= 85 and for F=90. 31. Plot to scale on the 80, 85, and 90 lines of your enlarged drawing the computed values of F^ these lines becoming 13, 14 and 15. Be careful to get each computed, F, on the proper vertical line. 32. Stretch a string, or place a ruler, along the points 1 to 15 of your drawing. Do the points added from your computed values seem to lie on the straight line through the points 1 to 12? With a ruler draw a single straight line through all the points of your drawing. 33. Mark a point on the horizontal line midway between the 35 and 40 points. What speed does this point represent? Draw a vertical from this point np to the line through the points 1 to 12. Mark the upper end of this vertical a. What force in pounds per ton does the line from a to 37^ represent? 34. How could you find from the drawing the number of pounds per ton needed to draw the train 42^^ mi. per hour? 67^ mi. per hour? 22^ mi. per hour? 15 mi. per hour? 17^ mi. per hour? 10 mi. per hour? 5 mi. per hour? 35. How could you find from the equation, i^= — 4-3, the numbers of pounds per ton needed to draw the train at the speeds 18, 36, 42, 57, and 69 mi. per hour? Compute these forces and compare them with the numbers found from measurements oi the drawing of Fig. 139. Remark.— The straight line through the points 1 to 12 is said to represent the equation, F=-^ -\- d. APPLICATIONS TO TRANSPORTATION PROBLEMS 269 §167. Laws for the Drawing of Loads. 1. The force (F), say 200 lb., needed to draw a load (L), say KOO lb., on a common road wagon over loose sand is given by the law F= iL, or F= .25Z. Find F for these loads. (1) Z = 864 1b.; (2) i = 1280 lb. ; (3) i = 2G48 1b.; (4) i = 32G8 lb. ; (5) Z = 4893 lb. 2. Over fresh earth, the law is F= .12^Z. What forces {F, in pounds) are needed to draw the following loads, in pounds, over fresh earth: (1) Z = 680? (2) i = 1624? (3) Z = 2160? (4) i = 3840? (5) Z' = 4580? 3. With common road vehicles on dry level highways the law of pulling (tractive) force (in pounds) h F= .025 i. L denotes the combined weight of the wagon and load in pounds. What forces will be needed to draw the following loads : (1) 45 bu. wheat on a 1200-lb. wagon? (2) 2 T. coal on a 2200- Ib. wagon? (3) 1^ T. hay on a 1680-lb. wagon? (4) A traction engine weighing 8^ T.? (5) A thresher, weighing 4^ T. 4. On well packed gravel roads the law is F= .052 L. To draw a certain load on gravel road a tractive force of 44.72 lb. was exerted. What was the load L? Solution. — Dividing both sides of the equation by .052, we have IT IT -7— = i, or, what is the same thing, L = -T^p^r. We have then, by sub- 44 72 stituting, L = ^^~- = 860. Ana. L = 860 lb. 5. Under the conditions expressed in problem 4, find the loads the following forces will draw : (1) i^.= 54.08 lb.; (2) /'= 66.56 lb.; (3) 7?^= 137.8 lb.; (4) 201.76 1b.; (5) 234 pounds. 6. The law of tractive force, on good, straight, level railroad track is ^= .0035 Z; on fair track, straight and level, it is F = .0059 L. Weights of cars are givefl in problems 15 and 17, p. 129, and in problem 19, p. 2G6. Make and solve problems based on these facts. 270 RATIONAL GRAMMAR SCHOOL ARITHMETIC 7. The tractive force (F) that can be exerted by a locomotive on dry track, straight and level, is about .3 of the part of the weight of the locomotive which rests on the drivers. The loads that c^n just be moved along by a locomotive are given by the equations of the last problem. Make and solve problems based on the following actual weights upon the drivers of certain loco- motives: (1) 80,890 lb. ; (2)*feo,850 lb. ; (3) 86,030 lb. ; (4) 106,875 lb. ; (5) 112,190 1b.; (6) 131,225 1b.; (7) 14:1,320 1b.; (8) 202,232 lb. (See also Fig. 139). 8. Problems may also be made on the following laws for road wagons: (1) Broken stone (fair) . . ii^=.028 i; (2) Broken stone (good) . • jP=.015 L; (3) Worn macadam . . . . jP=.033i; (4) Nicholson pavement . . F= .01^9 L; (5) Asphalt pavement . . . F^.012 L; (6) Stone pavement . , . F=.019 L; (7) Granite pavement . . . jP=.008i; (8) Plank road . . . . . F^.OIOL. 9, Find the value oif a; in these equations : (1) .35a; = 7 ; (2) .58a; = 11.6 ; (3) 1.28a; = 3.84 ; (4) .092a; = 3.68 ; (5) .019a; = 133; (6) 7.51a; = 302.8; (7) •175a; = 10.5; (8) 1.093a; = 13.116. 7 Solution of (1): a? =-5-= 20 .oD 10. Writing law (1) of problem 8 in the form .028Z = F, find the loads (L) to 2 decimals, the following, forces (F) will draw under the conditions of problem 8 (1) : (1) 50 lb. ; (2) 135 lb. ; (3) 56.28 lb. (4) 165i lb. ; (5) 68.58 lb. ; (6) 110.35 lb. 11. To find the loads (i) th&t given forces {F) will draw, the laws of problem 8 are more conveniently written^thus : L = ,j^^^ F= 35.71 F. Change other laws of problem 8 to this form. 13, , Fii idtl le load, of^: (1) i?'= 66 1b. (2) F^ 363 lb. 14, . Fiud L in F= (1) F= 120 lb. (2) F= 600 lb. APPLICATIOIS^S TO TRANSPORTATION PIIOBLEMS 271 12. Find the load, />, in Z = 35.71 F, for each of the following values of the force F: (1) i^=2801b.; (S) F= 78.6 1b.; (5) i^= UO lb. ; (2) i^=145lb.; (4) i^= 175.8.1b. ; (0) /^= 700 lb. . 1 F* Ly in L = -xg^ FyOT L= a, for these values ; (3) F^ 132 lb. ; (5) F= 26.4 lb. ; ; (4) F= 528 lb. ; (6) F= 52.8 lb. = .012 L for the following values of F. ; (3) F= 720 lb. ; (o*) F = 840 lb. ; ; (4) i^= 480 lb. ; ((>) E = 84.48 lb. 15. Find a; in y = .325 x for each of the following values of y: (1) y = 65; (4) ?/ = 29.25; (7) 7^ = 526.5; (2) y = 97.5; (5) y = 58.5; (8) y= 10.53; (3)*y = 48.65; (6) y = 17.55; (9) ?/ = 357.5. 16. On fair track, straight and level, the force which an engine can exert to pull a load is about j*^ of that part of its weight which is carried by the drivers. Show that this may be written as an equation thus : 4 F=^ — W. 15 Tell what the letters stand for. 17. By the aid of the law F-^ ^^ W, find the pulling force of engines having the following weights on the driving wheels: (1) r= 105000 lb.; (4) W= 99000 1b.; (7) W= 87300 1b.; (2) W^ 90000 1b.; (5) W^= 107200 lb. ; (8) }r= 171300 lb. ; (3) }r= 101500 lb.; (6) .PF= 121500 lb. ; (9) ir= 136350 lb. 18. Show how to change F=^^ ir first into 15 F= 4 irand then into Tr= 3.75 F, The letters mean the same here as in prob. 12. State the law W= 3.75 2^ in toords, 16. Find the weights on the driving wheels of engine^ that can exert the following pulling forces: (1) i^= 28000 lb. ; (3) F= 26400 lb. ; (5) 37600 lb. ; (2) F= 30400 lb. ; (4) F= 32640 lb ; (6) 39640 lb. 20. Find the values y in the equations if x = 138.5 : (1) y = 32 3;; (3) y = .036.r; (5) z/ = .0036a;; (2) y = 16.8 x; (4) y = .065 x\ (6) y = .0159 x. 372 RATIONAL GRAMMAR SCHOOL ARITHMETIC CONSTRUCTIVE GEOMETEY §168. Problems. Problem I. — Draw a perpendicular to a given line from a point upon the given line. Explanation.— Let AB denote the given line, and let P be a point upon AB. We wish to construct a perpendicular to AB at P. Place the pin foot on P and close the compass feet until their distance apart is less than either PA or PB. Pi is such 3^4 a distance. Then with Pi as radius draw a short arc across AB at both 1 and ^. Now spread the feet a little wider than Pi, place the pin foot on 1 and draw arc S. Arc S may a be drawn either above or below P. P 12 ^ In Fig. 140 it is above P. With- FiGUBB 140 ^^^ change of radius put the pin foot on 2 and draw arc 4 across arc 3. Let D be the intersection of arcs S and 4- With the straight edge of your ruler draw the line PD, PD is the desired perpendicular to AB, 'A* + EXERCISES 1. Draw straight lines on your paper, mark a point upon each line and draw perpendiculars through the marked points until you understand fully how it is done. 2. Draw the perpendicular to any line you may draw on your paper through a marked point on the Jine, drawing arcs 3 and -4 below the line. 3. Draw a straight line on the blackboard, mark a point upon it, and with crayon, string, and ruler draw a perpendicular to the linor through the marked point. 4. Draw a straight line on the ground and along the edge of a board, mark a point on it, and with cord, stake, and a straight board, draw a perpendicular through the point. Problem XL — Draw a perpendicular to a given line through a point outside of (not on) the given line. COKSTRtTCTlVE (^EOMETHY 2^3 Explanation.— Let AB denote the given line and P the marked point outside of AB. We wish to draw a perpendicular to AB through P. Put the pin foot on P and „ spread the feet far enough apart so that the pencil foot will reach helow the line AB. Then draw a short arc at 1 and at ^. Put the pin foot on 1 and draw arc S. Then with pin foot on $f draw arc 4 crossing 3 at D. Connect P and D with ruler and pencil. PD is the desired perpendic- ular. NoTE.^The arcs S and 4 might be drawn with radii of any length greater than half the distance from 1 tx) S; but both S and 4 must be drawn with the same radius. "^ -^^ ;c^ FlGUBB 141 EXERCISES 1. Draw straight lines on paper or on the blackboard, mark points outside of them, and draw perpendiculars to the lines, until the way of doing it is fully understood. 2. With cord, stake, and edge of a board draw a perpendicular on the grotind through a point not on a line. (Fig. 37, p. 101). Problem III. — To draw a square inside of a circle of radius |^". C Explanation.— Let 0, Fig. 142, be the center and let OB = }i". Draw a diameter AB. Spread the compass feet apart wider than {i" and putting the pin foot first on A, \B then on B, draw the arcs 1 and 2. Througli E and O draw a line and extend it both upward and downward until it cuts the circle at C and D. How do AB and CD comparo in length? With ruler and pencil connect Band C; C and A ; A and D; D and B. The figure BCAD is the desired square. exercises 1. Draw a square inside of a circle of 1" radius; of IJ" diameter; of 1|" diameter. 274 BATIOKAt OSAITMAB flCHOOl AMTttltmc 8169. To Model a S-Inch Cabe Hall Fiir 1^^ «y 'n««pendeiice in i'o**'^*?-^™® <?»y8t«l8 found form C^nlr ^'" ^ ^-^^ tenors ^^vr'^p--'-' -pat- pendacular^ rf^. «« in ProK ^iRi ' ?; l^-' ,*''■ Problem VI p. 181. On this line set off 3-ii,ch spaces and complete the deverif figure. Leave flaps as inSS PlODBB MS Old Independence Hall, PMladelphto ment, as shown in the tigure. Crease the paper along the lines over the edge of a ruler and fold up a 3- Seluherr^"^'^^^^^^'^*^"^ 1. What is the area of each sur- face? of all the surfaces? 2. How many edges has the cube? How many faxjes meet in each edge? The surfaces of the cube meet Cat'tU".*'*^'^"^'^™''^"-*. op,-, "a ■ P^™^" ^«r(ices (ver'-tl- ses). A single corner is a wrL. 3. How many vertices has the cube? How many edges meet at each vertex? The edges meet each other in <he corners of the cube forming ;o/Sr' 4. In how many faces does each vertex lie? i)oLs?r;oLr?^^"^*^^^^-^^'^^'^^«^-^ Figure 144 Development of 9f cube I>o surfaces? COKSTHUCTIVE GEOMETfiY 2'J'5 6. What are the limits of cubes? of surfaces? of lines'? Patterns of the surfaces of figures like those of Fig. 144 will hereafter be called developments, 7. Place an inch cube in the corner of the 3-inch cube. IIow many inch cubes will fill the row along one edge of the larger cube? 8. How many such rows will form a layer on the bottom? 9. How many such layers will fill the cube? 10. How many inch cubes will fill a 3 -inch cube? 11. How many cubic inches are there in a 4-inch cube? in a 10-inch cube? in an a;-inch cube? §170. To Model a Square Prism. .d ^^^^h Draw a straight line ad (Fig. /^ 1 45) 12 inches long, and make ah and ay dc each 2 inches long. Draw en and fo perpendicular to ad at b and c. Lay off distances eS, hg^ gl^ In^fc^ chj km, mo^ each equal to 2 inches. Draw ef^jk^ etc., and complete the development. Provide flaps; cut and paste the model. • figubbhs Development of the Square Prtsm 1. What is the area of each of the end surfaces? 2. Give the area of each side surface; of all the side surfaces. 3. How many inch cubes will the model hold? 4. By measuring the edges how could you find the number of cubic inches the model will hold? 5. How many cubic inches would it hold if it were 1 inch longer? twice as long? 6. If every line in the pattern were twice as long as in the development, how would you answer questions 1 to 4? Pupils should make their own models, different pupils using different lengths of lines and even devising different forms of pattern. For example, let one pupil use a 3-inch fundamental distance instead of an inch, another a 4-inch distance, and so on, each comparing his completed model with other models. il6 HATIOKAL OBAltMAB 80S00L ABITHMETtC §171. To Model a Plat Prism. >fy ^ k I f 1 ! ; h n c d P r s !Z Figure 146 Development^of Flat.Prism Draw a line ae (Fig. 146) 18 in. long and mark off ed = d in., dc = 6 in., cb= 3 in., and ba = 6 inches. At c and d draw perpendiculars to ae and extend them 11 in. below c and d to r and «, and 3" above c and dtok and /. Make co and dp 8 in. long and draw mq through o and p. Complete the development by drawing the necessary parallels. Provide it with the necessary flaps and paste up the model, leaving the top ckdl open. 1. With the aid of the model of an inch cube, find how many cubic inches would fill the flat prism. 2. To what is the product of the 3 different edges equal? 3. How could you obtain the capacity in cubic inches of such a prism by measuring the lengths of its edges? 4. How many square inches in the whole surface of the flat prism of Fig. 146? §172. CompariBon of Prisms. (a) A right prism. Using the lengths of lines as shown in Fig. 147, draw a pattern on heavy paper and construct the model of a right prism as was done in Fig. 145. Provide the edges with flaps and paste up the model, leav- ing one end open, so that it may be filled with sand. How many cubic inches of sand will just flll the model? How can you find the capacity of the model by using the lengths of the edges? s" 3" s;^^ ^ 3" 3" FIOUBE 147 Development of Bfght Prism CONSTRUCTIVE GEOMETRY 277 (b) An oblique prism with parallelograms for bases. Similarly draw a pattern, cut, fold, and paste up a model of the oblique prism, as shown in Pig. 148, leaving an end unpasted for sand. When the model of the right prism is just full of sand, the sides not being bulged out with e^-?- the sand, pour the sand into the model of the oblique prism. Does it fill the second model? What is the ratio of the capac- ities of the two models? What is the ratio of the areas of the bases (ends)? (c) A triangular prism. X X a d / 3" 3" \a; /S" h c ^''X /^ X N r 3'* FiGTTBID 148 Derelopment of ParaUelogram Prism X -x 1^ r V' ^ . •^ 1'' y'/} ^^v V' 1'' // '*^ L-i:— : — .- ft Model a triangular prism like the one shown in Fig. 149. Compare the capacity of this model with that of the model of a square prism l"xl"x4". What IS their ratio? Model a triangular prism such as that of Fig. 149, but use for lengths 3" instead of 1" and 7" instead of 4". How does its capacity compare with that of the prism of Fig. 147? of Fig. 148? Give the ratio in each case. Compare the ratios of the bases. §173. Volume of an Oblique Prism. The volume of a figure is the number of cubical units in the space enclosed by its bounding surfaces. 1. How many cubic inches are there ^^ --:;:^' j in a straight pile of visiting cards 2" high, if figubb im each card is 2" x 3" ? ^^^ ^S ^'^''' Figure 149 Development of Triangular Prism f 278 RATIOKAL GRAMMAR SCHOOL ARITHMETIC 2. How many cubic inches would there be in the pile if it were 5" high? 9" high? a in. high? 3. Push the straight pile of problem 1 over as in Fig. 151. How many cubic inches of paper are there in this oblique pile? 4. Has the height of the pile been changed in Fig. 151? has the area of the base? has the volume? 5. How can you find the number of cubic units in a right prism from tbo Figure 151 area of its base and its height? in an Oblique Pile, or Oblique Prism oblique prism? 6. Find the volumes of square prisms having edges of the following lengths : (1) 3" X 5" X 6" ; (4) 4f ' x 6' x 12' ; (7) a in. x ^ in. x c in. ; (2) 6" X 5" X 9" ; (5) 16' x lOf x 20' ; (8) a; f t. x v ft. x ;2; f t. ; (3) 15"x9"x8"; (6) 45" x 16|" x 21"; i9)myd.xnyd.xpjd. §174. Paper-Folding. For the following exercises in paper-folding any moderately thick, glazed paper will do. Tinted or colored paper, without lines, will however show the creases more clearly. It is con- venient to have the paper cut into pieces, about 4" square. Such paper is inexpensive and may be had of any stationery dealer. Problem I. — At a chosen point on a line, make a perpendicular to the line, by folding paper. Explanation. — Fold one part of a piece of paper over upon the other and crease the paper along the fold, as at AB, by drawing the finger along the fold. Taking D to denote the chosen point, fold the paper over the point 2), and bring the two parts, DA and DB, of the crease AB exactly together. Hold the paper firmly in this position and crease the paper along the line DC. Compare the portion of the paper between the creases DB and DC with the portion between the creases DA and DC. How do the angles BDC and CD A compare in size? Definitions. — When two lines meet in this way making the angles at their point of meeting (inter- section) equal, the lines are said to be perpendicular to each other, and each is called a perpendicular to the other. The angles thus formed are called right angles. Figubb 152 COKSTRUCTIVE GEOMETRY 279 Problem II. — Bisect an angle, by folding paper. Figure 153 Explanation.— Crease two lines, as OA and OB, lying across each other. They make the angle AOB. Now fold the paper over, and bring the crease OA down on OB. Holding the creases firmly to- gether crease the bisector OC. How do the angles AOC and BOC compare in size? Do they fit? What is the ratio of angle AOC to angle BOC? of AOB to AOC? How could the angle AOB be divided by creases into 4 equal parts? Problem III. — Crease three non-parallel lines. Explanation. — Crease AB in any position. Crease CD in any posi- tion, not parallel to AB. Finally, crease EF in any position not parallel to either AB or CD, Fig. 154. In general, in how many points do three non-parallel lines cross each other? Can you crease three lines, no two of which are parallel, in such a way as to obtain just two crossing points (intersec- tions)? Try it. Crease three non-parallel lines in such positions as to give but one intersection (see Fig. 155). Lines which go through the same point are called concuTrrent lines. k ^ —!- r B l> iC F, A O B E: \D .1 FIOUBE 154 FIGUBB 155 Problem IV. — Crease the bisectors of the 3 angles of a triangle. Explanation. — Crease out a triangle such as ABC. Then crease the bisector of each angle, as in Problem II. Work carefully. 1. How do the bisectors cross each other? 2. Since the three bisectors of the three angles of a triangle all go through 0, what name would be applied to them Figure 150 280 RATIONAL GRAMMAR SCHOOL ARITHMETIC Problem V. — Bisect a given line, by paper-folding. j mmf r r- .^ |™- — , EXPLANATION. — Crease a line and stick the point ■ "" ' *^ of a pin through the crease at A and at B, AB is the line to be bisected. ^1 „| Fold the paper over and bring the pin hole at *.4^-»™--H.i; ,p-n.^™™nrrt3 B down on the pin hole at A. Press the paper down and crease a line across AB, as at C. CD is I the perpendicular bisector of AB. i ,„, lln „,„ niii J Crease a line from D to B and another from D to L - ^-" "^^^ ^ ^^"^ ' ^ '^^ A, When the paper is folded over the line CD, how •r^ — do these two creases seem to lie? Ck)mpare the FiQUBB IW lengths of DB and of DA. 1. Crease the perpendicular bisectors of the 3 sides of a triangle and find how they cross each other. 2. What kind of lines are the perpendicular bisectors of the sides of a triangle? Problem VI.— Crease a square and its diagonals. Explanation.— Crease 2 lines, as AB and AX, ' -X":' T perpendicular to each other. (See Problem L) Fold the paper over the point B and when the ^ ^ two parts of the crease AB fit, crease the line BY. q Now fold the paper over so that crease BY comes down along crease BA, and crease the line BC. _^ Fold the paper over a line through C, bringing -V, ^' \ CX down along CA, and crease CD. * — -^ L — J ABCD is the required square. FIGURE 168 1. When the square is folded over the diagonal BC, where does D fall? Along what crease does BD lie? CD? 2. How does the diagonal divide the area of the square? 3. Fold over and crease the diagonal AD? How does it divide the square? 4. Compare OB with DC; OD with OA. 5. How do the diagonals of a square divide each other? G. How do the diagonals divide the area of the square? 7. How does AD divide the angle BAC? the angle BDC? 8. Fold a second square and crease its diagonals (Fig. 159). Fold over 0, bringing D down on HO and crease JIG. Similarly crease IJF. OONSTBUCTIVB GEOMETRY 281 r - ■' "^ r ,f D £ o !• J ' 1 i t 1 9. When the paper is folded over OF along what line does ODtaM? OH? 10. How does OB compare with OD in length? How then do .4Z> :md ^(7 compare? 11. How do HF and HG divide the square? 12. Crease the following lines : GF^ FH^ HE^ and EG. How does the area of the square GFHE compare with that of ABDC? figurb 159 Problem VII. — Crease the three perpendiculars from the vertices of a triangle to the opposite sides. Explanation.-— Crease the triangle ABC, Fig. 160. Fold the paper over the vertex C, and bring the crease DB down along DA, and crease the per- pendicular CD. Similarly crease AE and BF, Work carefully. How do the creases CD, AE and BF cross each other? What kind of lines are the three ^rpen- diculars from the vertices to the opposite sides of a triangle? Figure ifto Problem VIII. — Crease the three perpen- dicular bisectors of the sides of a triangle, Fig. 161. How do they cross? What kind of lines are they? Problem IX. — Crease a rectangle and its diagonals. Explanation.— Crease AB, Fig. 162, making the distance from A to B two inches. By Problem I, crease the perpendiculars AD and BC. Make AD and BC each 1" long and crease CD. Crease the diagonals, one through A and C, and the other through B and D. Call their crossing point O. Fold the paper over the perpendicular FE, bring- ing B down on A. Where does C fall? What other line equals CB? OB? OC? Fold the paper over the point O so that B falls .^ on C, and crea.se the perpendicular HG. What FiGTOB 168 other line equals AB? OG? OD? How does the intersection, O, of the diagonals divide the diagonals? FIGUBB 161 ■J . '■ s.^ f^;i|w|| 282 RATIONAL GRAMMAR SCHOOL ARITHMETIC §175. Perimeters. — The perimeter of any figure is the sum of the lines bounding the figure. Thus, in form (0), Fig. 163, if ^ de- note the perimeter, (I) p = x + t/ + z. 1. What does x ■h2x + 4:X mean? IIow may it be more briefly written? 2. What is the coefiicient of x in the answer to problem 1? 3. Write the perimeter, p, of (1), Fig. 163, in two ways. 4. Write the perimeter, p, of (13) in three ways. 5. Write the perimeter, p^ of (12) in three ways. 6. Write i of the perimeter, p^ of (13) in two ways. 7. Write an expression showing that the two answers to 6 are equal. Figure 163 8. In (15), if a; = 80 rods and y = 40 rods, how many feet in the perimeter? 9. In (1) and (2), if x and y each = 20 rods and one side of the triangle rests on the square, making a new figure, omitting the common line, what is the perimeter of this new form, in feet? How many sides has the new figure thus formed? 10. Forms (1) and (13) are combined into a single figure. Write p for the new figure in two ways, supposing x the same in both. CONSTRUCTIVE GEOMETRY 283 §176. Quadrilaterals. 1. Forms (2), (7), (8), (10), (11), (12), and (13), Fig. 163, aro differBnt kinds of quadrilaterals. What is a quadrilateral 9 2. What quadrilaterals have their opposite sides parallel? These figures are parallelograms. Define a parallelogram, 3. What parallelograms have all their sides equal? What is a rhombusf 4. What rhombus has all its angles equal? What is a square? 5. What quadrilaterals have their opposite sides equal but con- secutive angles not equal? Define a rhomboid. 6. What parallelograms have their angles all equal? Define a rectangle. 7. What quadrilateral has only one pair of sides parallel? De- fine a trapezoid. 8. Is a trapezoid a parallelogram? Is it a quadrilateral? 9. Is a rectangle necessarily a quadrilateral? Is it a parallel- ogram? a square? May a rectangle be a square? 10. Is a square necessarily a quadrilateral? Is it a parallel- ogram? a rectangle? a rhombus? §177. Perimeters of Miscellaneous Figures. 1. Denote the perimeter of each of the forms in Fig. 163 by p and write an equation like (I) in §175, showing the value of p for each figure. 2. Omitting the lines on which the forms join, write an equa- tion showing the value oip when (2) and (6), Fig. 163, are joined on y, which has the same value in both. 3. In the same way join (3) and (10) on a, which has the same value in each, and write an equation showing the value of p. What is the name of the figure thus formed? 4. Join (2) and (7) in which x and y are equal. If the perimeter of the figure thus formed is 240 rods, find the value of x in feet. 5. What is the length of the perimeter of a figure like (6), Fig. 163, whose sides are 2 in., a in., and h in. long? whose sides are X in., 8 in., and z in. long? like (10), whose sides are c ft., 2c ft., V ft., and X ft.? like (12), whose sides aro 2a; rd., 4y rd., 2a: rd., and 4^ rd. long? 284 BATIONAL GRAMMAR SCHOOL ARITHMETIC Write the perimeter (j^) of each of the figures below. (i) Square Rectangle b c « (B) y ^ y y («) -x * „ -b— ^ t i a? a; X (7) i i 1/ y ^ —y— ^ h 6 a («) a X y y (9) d d UO) FiGUBB 164 In such forms as' those from (o) to (10), Fig. 164, some line, or lines, must be found by subtracting others. In (6) for example, note that the ends are each x — y. The perimeter is then a? + a; + (x — y) •\-y -\-y + {x- y) = 4a; — y — y+ y + y = 4a: — 2y+ 2y=4a?. All sides not lettered must be expressed without using other letters than those given on the figure. The perimeter means the sum of all the lines that bound the strip, or surface, of the figure. Note. — One-half the difference of a line m and a line n is written —^ — or! {yn — n), L In (8), a = 15', h = 12', x = 25' and v = 30'. Find the length of the perimeter of the figure. 2. Find the area of (8) enclosed by the solid lines. 3. Make and solve other similar problems. CONSTRUCTIVE GEOHETBY 285 §178. Heasnring Angles and Arcs. ORAL WORK We may measure the amount of turning of each clock hand in either of two ways : (1) By the length of the circular arc passed over by the tip of the rotating hand; (2) By the wedge-shaped space having its point at the hand-post, A, over which the stem, A-III, A-VI, etc., of the rotating hand moves. While the clock hand turns from XII to III its tip moves over 1 quadrant of arc FiGURB 165 ^^^ j^g ^^^^ moves over a right angle. 1. While the hand turns from XII to VI over how many quadrants does its tip move? Over how many right angles does its stem turn? 2. Answer same questions for a turn of the hand from XII to IX ; from XII to XII again ; from XII around through XII to III. 3. Over what part of a right angle does the stem of the hand move while the hand is passing from XII to I? from XII to II? from III to IV? from VI to VIII? 4. Over what part of a quadrant does the tip of the hand pass in each case of problem 3? 5. Over what part of a right angle does the stem of the hand move while the tip moves 1 min. along the arc? In the same case over what part of a quadrant does the tip move? Definitions. — Any wedge-shaped part of the face moved over by the stem of a clock hand as it turns around the hand -post is called an angle. The curve passed over by the tip of the hand, while the stem of the hand moves over the angle, is called the arc of the angle. Illustrations. — The wedge-shaped spaces, having their points at A, and included between any two positions of the hand, as A-XII and A-I, A-1 and A-III, A-I and A-VI, are all angles; the space swept over by the hand, A-I, as it moves around through II, III, IV, V, VI, etc., to IX is also an angle. 6. As the hand moves from A-XII to A-VI, that is, so that the two positions of the hand are in the same straight line, the hand moves over a straight angle, A straight angle equals how many right angles? 280 RATIOKAL GRAMMAR SCHOOL ARITHMETIC 7. The arc of a straight angle equals how many quadrants? How many quadrants make a complete circle? 8. How many 5-min. spaces make the circumference of a complete circle? How many 1-min. spaces make the circumfer- ence of a complete circle? 9. What part of a right angle is passed over by the stem of the hand as its tip moves from one end to the other of a 1-min. space? 10. If lines were drawn from the hand-post to the ends of all the 1-min. spaces, the whole face of the clock would be divided up into how many small angles? The instrument in common use for meas- uring angles and arcs is the protractor. (See Fig. IGG.) 11. Instead of divid- ing the right angle up by radiating lines into 15 equal parts, the protractor divides the right angle into 90 equal parts, one of which is called the angular degree. These same lines would divide the quadrant up into how many equal parts? Each of these parts is a degree of arc. 12. How many angular degrees are there in a straight angle? How many degrees of arc in the arc of a straight angle? 13. How many angular degrees are swept over by a clock hand while moving entirely around once? How many degrees of arc in the circumference of a circle? 14. How many degrees of angle are passed over by the stem of the minute hand in two hours? In the same time how many degrees of arc are passed over by the tip of the hand? 15. A sextant is ^ of a circumference; how many degrees of arc are there in a sextant? 16. An octant is ^ of a circumference; how many degrees of arc are there in an octant? 17. Study the protractor, Figs. 166 and 167; notice how its marks are numbered. How many degrees of arc are there between Figure 166 COlfSTRUCTIVE GEOMETRY 287 the closest lines on the outer edge? How many degrees of angle are there between the lines which converge toward the center? 18. For smaller angles a shorter unit is 75V <^^ ^ degree of angle and the smaller unit is called the minute of angle. Vir ^^ ^^^ Figure 167 minute, called the second^ is a still smaller unit. How many minutes of angle in a right angle? in a straight angle? in a com- plete revolution, or perigon? How many seconds, in each case? 19. The arcs between the sides of the minute and the second angles are the minute and the second of arc. How many minutes of arc in a quadrant, in a sextant, in an octant, in a circumfer- ence? How many seconds, in each case? A paper protractor can be purchased at any bookstore, and each pupil should supply himself with one. To measure an angle the center of the protractor is placed on the vertex A, and the line through A is placed along one side AD (see Fig. 168). The reading of the mark below which the other FlGUBE 168 side, AF, falls is the number of degrees in the angle, or in the arc of the protractor included between AD and AF. 288 EATIONAL GRAMMAR SCHOOL ARITHMETIO WRITTEN WORK 20. Draw a triangle, and, with a protractor, measure its angle*. To what is the sum of all three equal? 21. With ruler draw a half a dozen triangles of different shapes, carefully measure each angle, and find the sum of tlie three for each triangle. 22. Measure with the protractor the number of degrees in the angle at one comer of the page of this book. 23. Draw two straight lines crossing each other as in Fig. 160. With the protractor measure and compare a and c\ b and d» The angles a and c, or b and rf, lying opposite each other, are called opposite angles^ or vertical FiGUBB 169 atigles. 24. Draw two crossing lines in different positions, and in each case compare the measures of a pair of opposite angles. What do you find to be true? 25. Draw a pair of parallel lines, as a and J, Fig. 170, and draw a third straight line c cutting across the parallels. Measure with a protractor and compare the •- four angles, in which 1 is written in Fig. 170. Measure and compare those in which 2 is written. What do . _ f , , Q FlGUBB 170 you find to be true? 26. Draw a line perpendicular to another, see §168, and measure all the angles formed. What do you find?* Definitions. — ^The lines which include the angle ^^yq sides of the angle. The point where the sides meet is the vertex of the angle. 27. Draw a quadrilateral of any irregular shape and measuro each of its 4 angles. To what is the sum equal? Try another quadrilateral and see whether you obtain the same sum. 28. Draw a given angle at a point on a given line. Explanation.— Let the given line be ED, the given angle 35", and the given point A, Fig. 168. Place the protractor with its center at A, and with the diameter of the protractor along the line ED. Make a dot opposite the 35" mark on the protractor. With the ruler draw a straight line through A and this dot to F. The angle DAF is the required angle. 29. Draw a straight line, mark a point on the line and draw the angle 76°; 95°; 100°; 135°; 55°30'. ifijEr COK8TR0CTI7'E aEOMETRY 289 Table op Units of Angle and Arc Measurement 60 seconds (") = 1 minute (') 60 minutes = 1 degree (**) 860 degrees = 1 circumference (or perigon) 90 degrees = 1 quadrant (or right angle) Table op Equivalents 1,296,000" 21,600' 360** 4 quadrants = 1 circle §179. The Sum and Difference of Angles. Exercise 1. — With ruler and compasses, draw AB and CD perpendicular to each other. 1. Suppose a protractor supplied with a pointer as shown in Fig. 171, the center of the protractor being placed at the point, 0, over how many degrees of arc would the tip of the pointer turn while the stem of the hand turns from line OB to line 0(7? from OB through OCU) OAl 2. How many degrees are there in the angle jSOCT in angle BOAl in COD'i in A on? in the angle from OA around through OD to OB? from OA around to 00? from OA entirely around to OA again? 3. How many degrees of angle fill the space around a point, as 0, on one side of a straight line, as AB? on both sides? Definitions. — An angle that Is smaller than a right angle is called an acute angle (see Fig. 172). An angle that is larger than a right angle is called an obtuse angle. Acute Angles Obtuse Angles FXOURB 173 200 RATIONAL OHAMMAK SCHOOL ARITHMETIO Figure 173 Exercise 2. — Find the sum of two angles 1. Draw two acute angles like those at the top of Fig. 173,care- j^ f iilly cut them out, and place them as ^^'^ in the lower part of the figure. Posh their vertices and nearer sides en- tirely together. Draw two lines along the other sides of the angle. Bemoye the angles and have an angle like that on the left, which is the sum of the two angles. 2. Calling the larger angle re, and the smaller y, what denote the last angle? 3. Draw an angle equal to the sum of an acute and an obtnae angle. Exercise 3. — Find the difference of two angles. 1. Draw two angles like those at the top of Fig. 174, cut them out, and place them as in the lower, left-hand part of the figure. Mark along the lower side of the upper angle and cut along the mark. The lower part of the larger angle (shown on the right) equals the difference of the two angles. FIGDRB 174 Figure 175 Figure 176 2. If the larger angle equals x degrees and the smaller equals y degrees, how many degrees are there in the difference? CONSTECCTIVE 6E0METBT 291 Figure 177 Definitions. — ^Two angles whose sum equals a right angle, or 90°, are called complemental angles (Fig. 175). Two angles whose sum equals 2 right angles, or 180'', are called sup- plemental angles (Fig. 176). The sum of all the angles that just cover the plane on one side of a straight line is equal to how many right angles (Fig. 177)? To how many degrees is this sum equal? Exercise 4. — Draw an equilateral triangle (Problem VII, p. 102). Tear off a corner and fit it over each of the other corners in turn. How do the three angles compare in size. Tear off the other corners and place them as in Fig. 178. To what is the sum of all three angles equal? If one of the angles is x degrees, how many degrees are there in the sum of all three angles? As the sum of the three angles rs>^ Figure 179 Figure 178 equals both dx degrees and also 180°, we may write the equation 3x = 180. If Sx = 180, to what is x equal? How many degrees are there in one of the angles of an equilateral triangle? Exercise 5. — Draw any scalene triangle. Tear off the corners and place them as in Fig. 179. To how many right angles is the sum of all the angles of the triangle equal? To how many degrees? 1. Letting a?, y, and z denote the numbers of degrees in the respective angles, in what other way may we write the sum of all three? What equation piay we then write? Tell what the equation x-{-y + z = 1S0° means. 2. Draw other scalene triangles, tear off the corners, place them as in the figure, and find whether x + y + z = 180° in all cases. 202 RATIONAL GRAMMAR SCHOOL ARITHMETIC 3, Crease a right triangle and find whether the equation, x-{-y + z = 180°, is true for it. 4. Cut along the creases and tear off the two acute angles of a carefully creased right triangle and fit them over a carefully creased right angle? What seems to be true? If this is true what equation may you write for the sum of the two acute angles {x and y) of a right triangle? Exercise 6. — Draw a quadrilateral and cut it along a straight line from one corner to the opposite comer as in Fig. 180. Such a line as is indicated by the cut is called a diagonal. 1. The cut divides the quadrilateral ^"^-*--.. — into figures of what shape? J~^y^ ^^.^"''^ ^- To what is the sum of the three ly* e/ l^__«_2/ angles of each part equal? FIGURE 180 3. To what is the sum of all six angles of the two triangles equal? 4. To what is the sum of all 4 angles of the quadrilateral equal? 5. Do your answers hold true for the parallelogram of Fig. 180? 6. Do they hoi d good for any shape of quadrilateral you can draw? 7. To what then is the sum of the four angles, x^ y, z^ and w, of any quadrilateral equal? Exercise 7. — Draw a hexagon and cut it along diagonals as •shown in Fig. 181. . 1. Into figures of what shape is the hexagon divided? 2. Into how many such figures do the diagonal cuts divide the hexagon? 3. To how many degrees is the sum of all the angles of all the triangles equal? 4. To how many right angles is the sum of all the angles of the hexagon equal? 5. Do all your answers hold true for the regular hexagon (sides equal and angles equal) of Fig. 181 also? 6. In each of these hexagons how many less triangles than sides of the uncut hexagons are there? Figure 181 CONSTBUOTIVE GEOMETRY 293 7. Draw an S-sided figure and find how many triangles the diagonal cuts from any vertex would give. 8. How many less triangles than sides wei:e given by the quadrilaterals of Fig. 180? 9. To how many right angles is the sum of all the angles of any 15-sided figure equal? of an w-sided figure? 10. The angles of a regular hexagon are all equal. How many degrees does each contain? Exercise 8.— Draw a rectangle and cut it out. Cut it along a diagonal, and denote the angles by letters, as shown in Fig. 182. 1. Turn the lower right-hand triangle around and place it on the upper piece so that e may fall at a, / at c^ and d at J. Can you make the parts fit? 2. How does a diagonal divide a rectangle? 3. Show the relative length of sides m and n by an equation ; of h and l\ of the angles a and e\ c and /; b and d; b + e and a + d. 4. From these equations point out those which show that the opposite sides of a rectangle are equal. 5. Show from your equations how the opposite angles compare in size. 6. With parallel rulers draw a parallelogram and cut it as sug- gested by Fig. .182. Answer questions 1-5 for the parallelogram. 7. Write the perimeters of the parallelogram and of the rect- angle of Figs. 180 and 182. §180. Products of Sums and Differences of Lines. 1. What is the area of A? of B? of C? Note. — The product of a and x-^y is writ- ten a(X'\-y) and is read **a times the sum ^- + 2/." ^ •*■ y 2. Read the equation ax + ay ^a (x + t/) FiGURai83 ^^^ explain its meanmg from Fig. 183. dp Draw a figure and show that axi-ay + az^a (a; -f y + 2;). X « a A B I Q c 1 294 RATIONAL GRAMMAR SCHOOL ARITHMETIC written 4. What is the area of each of the parts of Fig. 184? Note, a-^b times x-^y is I fp] 5. From Fig. 184 show the meaning of ^ each product in the equ ation {a + b) {x + y) = ax-j-ay + bx + by. Also show from Fig. x + y FIGUBB 18i 184 why the equation is true. 6. What is the area of each part of Fig. 185? 7. Write an equation showing how to mul- tiply x + y hj itself, or how to square x + y. 8. Point out from Fig. 185 the meaning of {x + yy=^a^ + 2xy + y^. 9. Point out from Fig. 186 the meaning of b {a — c) =ab -'be. x+y [SSI X y FiGUBS 186 t t c ' o 1 i C3 b PiGi: fBB] b L86 o+b FIGUI IE 18 - a -• 7 a -> ::::io 10. Draw a figure and show that b {a + cl — c) =ab + bd — be. 11, Show from Fig. 187 that {a + b){a-b)=a^-h'. -^-^ 12. Show from Fig. 188 that Figure 188 (a -b) (a-b) =a^ — 2ab + J*. 13. What is the area of the rectangle A BCD (Fig. 189)? of 5? 14. What is the length of the dotted part of BC? 15. What is the area of B? IG. What is the area of the cross-ruled part? 17. Write the areas of the cross-lined figures below (Fig. 190). Figure 189 COKSTRUCTIVE CJEOMETRY 295 Call the difference of the two areas d in each case, and answei with an equation. il) (») (3) (4) FiGUBS 190 18. A lot, 50' X 176', JB occupied by a house' which, with its porch, covers a 40' square. A cement walk 3^ wide runri from the front line of the lot to the porch, and from the back door of the house to the rear line of the lot. The rest of the lot is covered with grass. How many square feet of grass are there? 19. A man cuts an 18" strip of grass entirely around a rectan- gular lawn, 75' x 175' ; how many square feet of grass does he cut? 20. He then cuts another 18" strip around just inside of the strip mentioned in problem 19; how many square feet of grass does he cut the second time round? 21. A farmer reaps a 12' swath entirely round a rectangular wheat-field that is 40 rd. x 80 rd. How many square rods does he reap the first time round. (12' = y\ rd.) 22. He again reaps a 12' swath entirely round the field. How many square rods does he reap the second round? the third round? 23. A farmer plows an 18" strip entirely round a rectangular piece of ground 20 rd. x 80 rd. How many sq. ft. does he plow? 24. He continues plowing round the rectangle (prob. 23) until he has a strip plowed 1 rd. wide entirely round it. How many acres has he then plowed? 25. How many acres has he plowed when the plowed strip is 2 rd. wide? 4 rd. wide? 26. How many acres has he plowed when the unplowed strip of prob. 23 is 4 rd. wide, the plowed strip being of the same width all round the rectangle? 27. How many acres have been plowed when the unplowed strip remaining is 3^ rd. wide? 2^ rd. wide? 296 BAtlONAL QEAMMAft SCHOOL ARlTHMEtlO PLACES ON THE EARTH 297 §181. Locating and Describing Places on the Earth. Fig. 191 is a map of the eastern and the western hemispheres of the earth,, showing the lines that are used for locating and describing places on the earth. An imaginary line, called the equator^ runs entirely round the earth, dividing its surface into the northern and the southern hemispheres. This equator is shown on the map by the heavy horizontal line. The heavy curved line going through the north and south poles of the eastern half of the map and crossing the equator at 0° is called the^rime meridian, ORAL WORK 1. What do the numbers 15°, 30°, 45°, etc., mean, that stand along the equator to the right and left of 0°? 2. What part of the map is north? south? east? west? 3. How many degrees are there between the curves that run up and down on the map through the poles? 4. How many degrees are there between the curves that run across the map from right to left? 5. Show the place on the map that is 15° (meaning 15 degrees) west of the zero (0°) point; 30° (30 degrees) west of 0° ; 60° W. ; 105° W. ; 150° W. 6. Show the place that is 15° east of the 0° point; 45° E. ; 60° E.; 90° E.; 120° E.; 150" E. ; 165° E. 7. Point to the place that is 15° north of 0°; 30° N.; 30° south; 45° S.; 60° N. ; 75° S.;.90° N. ; 90° S. 8. Show the place that is 30° W. and 15° X. ; 30° W. and 30° N.; 30° W. and 15° S.; 60° W. and 30° S.; 60° E. and 15° N.; 60° E. and 30° S. 9. Point out the places that have the following positions: (1) 30° W. and 75° N.; (8) 105° W. and 30° N.; (2) 30° W. and 30° S. ; (9) 60° W. and 60" S. ; (3) 45° E. and 15° S. ; (10) 105° W. and 45° S. ; (4) 60° E. and 45° N. ; (11) 90° E. and 30° N. ; (5) 45° W. and 45° N. ; (12) 90° E. and 90° N. ; (6) 45° W. and 45° S. ; (13) 180° W. and 75° S.; (7) 45° E. and 45° S. ; (14) 90° W. and 90° N. 298 RATIONAL GRAMMAR SCHOOL ARITHMETIC Degrees measured toward the west and toward the north are written with a plus (+) sign hefore them. Degrees measured toward the east and toward the south are nmrked with a minus (— ) sign before them. 10. Point out the following places : (1) +15°, +30° (2) +15°, -30° (3) +60°, -75° (4) -60°, -75° (5) -75°, +60° (6) -75°, -60° (7) +100°, -45° (8) +100°, +75° (9) +160°, +80° (10) - 40°, +65° (11) -140°, -35° (12) -120°, -65° WRITTEN WORK 1. Write down about the correct distances in degrees westward or eastward, and northward or southward, of the mouth of the Amazon Eiver; of the Mississippi River; of the Nile River; of the Ohio River; of the Niger River (Africa). 2. Write about the positions of these places, using + or — signs: Washington, D. C. ; New York City; Boston; Chicago; Denver; New Orleans; London; Paris; Berlin; the south point of Florida; the south point of South America; the east point of South America. 3. About how many degrees long is South America? Africa? The Nile River? 4. About how many degrees wide is South America? Africa? The United States? The curved lines running from right to left are called parallels; those running from north to south are meridians, 5. About how many degrees wide is the United States along the 30° parallel? along the 45° parallel? 6. About how many degrees wide is South America along the equator? along the -30° parallel? the -15° parallel? 7. Answer similar questions for Africa. 8. How many degrees are there between the prime meridian and the meridian of the mouth of the Nile River? between the prime meridian and the meridian of the southeast point of Arabia? of the southwest point of Arabia? PLACES ON THE EARTH 299 9. How many degrees are there between the 45th meridian and the meridian of the mouth of the Mississippi Eiver? 10. Chicago is about how many degrees west of New York City, that is, about how many degrees are there between a meridian through New York City and a meridian through Chicago? 11. About how many degrees is Chicago north of New Orleans? of Panama? of Rio de Janeiro? of the mouth of the Amazon River? 12. About how many degrees west from New York City is Denver? San Francisco? The mouth of the Ohio River? Distances measured westward or eastward in degrees along a parallel from the prime meridian are called longitudes. Distances measured in degrees along a meridian northward or southward are called latitudes, 13. About what is the difference between the longitude of New York and of Chicago? Boston and Denver? Philadelphia and New Orleans? New Orleans and San Francisco? Washington and Savannah? 14. Give the differences of latitude of the same pairs of places. The latitude and longitude in degrees of some important sea- ports are given here : PliACB LATITITDB LONGITITDB Cape Town . . . —33.93 — 18.47 Hamburg .... +53.55 — 9.97 Honolulu +21.30 +157.85 Manila +14.67 —121.00 Panama + 9.00 + 79.52 Quebec +46.80 + 71.32 Latitttdb Longitxtde . +51.85 + 8.28 Place Queenstown San Francisco. +37.78 +122.43 Savannah +32.02 + 81.12 Stockholm .... +59.38 + 18.05 Valparaiso —33.00 + 71.63 Yokohama .... +35.50 —139.58 15. What is the difference of latitude of San Francisco and Honolulu? The difference of longitude? 16. What are the differences of latitude and of longitude of San Francisco and Yokohama? of San Francisco and Valparaiso? of Hamburg and Savannah? of Queenstown and Quebec? of Cape Town and Stockholm? of Panama and Manila? of Panama and Valparaiso? of Manila and Yokohama? of Quebec and Panama? 17. Other problems may be made from the table. 18. Read the differences of latitude and of longitude of any places given on maps in your geographies. 300 RATIONAL GRAMMAR SCHOOL ARITHMETIC §182. Longitude and Time. The earth may be regarded as an immense sphere turning, like a top, round one of its diameters as an axis from west to east once every 24 hr. This carries the surface of the earth and all objects fixed upon it (as a schoolhouse) round through 360"" in 24 hours. Measuring one complete turn (rotation) of the earth in degrees, it may be said to be equivalent to 360°; measuring it in time, it may be said to be equivalent to 24 hours. This gives us the following tables of equivalent measures: 360° correspond to 24 hr. ; 1° corresponds to ^^^ of 24 hr. = -^^ hr. = 4 min. of time; 1' corresponds to -^ ot 4 min. = ^^ min. = 4 sec. of time ; 1" corresponds to -^^ of 4 sec. = j^ s®^- ^^ time. 24 hr. correspond to 360° ; 1 hr. corresponds to 15° ; 1 min. of time corresponds to -^ of 15° = J** = 15'; 1 sec. of time corresponds to ^ of 15' = i' = 15". All objects seen on the sky, as the' sun, may be regarded as stationary, while the turning of the earth carries us past them from the west toward the east. This makes the sun appear to rise in the east, move over, and set in the west. 1. Will the sun pass over eastern or western places earlier? Which places have later local* times, those over which the sun passes earlier, or later? Which places have earlier local times, eastern places, or western places? 2. What time is it at a place 45° west of Washington when it is 10 o'clock at Washington? At the same instant, what time is it at a place 30° east of Washington? Definition. — Longitude is the distance in degrees, minutes and seconds (of arc) due eastward or westward from a chosen meridian, called the prime meridian. Astronomers and navigators have agreed that the prime meridian shall be the meridian of the Royal Observatory at Green- wich, England. 3. The difference between the local times of two places is 4hr. 3 min. ; what is the difference in longitude between them? * Local sun time Is obtained by setting timepieces at XII as tbe sun crosses the meridian. LONGITUDE AND TIME 301 4. The difference of longitude between two places is 105° 45'; what is the difference of their local times? 5. It is 4: 50 (4 hr. 50 min.) p.m. at a certain place and 1 : 48 p.m. at another; which place is east of the other and what is the difference of their longitudes? 6. Two men met in Chicago with their watches keeping the correct local times of the places whence they came. On comparing their times one watch showed 10:47 p.m., and the other 3:18 p.m., when it was 9: 26 p.m. in Chicago. From which direction did each man come? 7. If the watches (problem 6) were keeping the correct local times of their places, what were the differences of longitude between the places and Chicago? 8. Explain how it happened that the announcement of Queen Victoria's death was read in the Chicago dailies at an earlier hour than that borne by the announcement itself. 9. How may it happen that a cable message sent Wednesday forenoon may be received at a remote place Tuesday? 10. The local times of two ships at sea differ by 4 hr. 18 min. 15.6 sec. ; what is the difference of their longitudes? 11. One ship is in longitude 186° 40' 12" west and another is in longitude 20° 16' 48" west; what is the difference of their longi- tudes? of their times? 12. The longitude of one ship is 3° 28' 10" west and that of another is 18' 38" east. What is the difference of their longitudes? of their times? 13. The longitude of the Observatory of Madrid, Spain, is 3° 41' 17" west and that of the Berlin Observatory is 13° 23' 43" east. What is the difference of their longitudes? of their times? 14. The longitudes of the Cambridge (Eng.) and of the Paris Observatories are 5' 41.25" east and 2° 21' 14.55" east respec- tively. What are the differences of their longitudes? of their times? 15. When the sun is on your meridian, that is, when it is noon at your place, at what places on the earth is it forenoon? after- noon? night? midnight? 302 RATIONAL GRAMMAR SCHOOL ARITHMETIC The longitudes from Greenwich, Eng., of places are also often given (as below) in hours, minutes and seconds of time. The plus (+) sign means that the place is west of the meridian of Greenwich, and the minus (— ), that the place is east of this meridian. j^ Ar™ liONGITITDB TBOM «- .^-_ LONGITUDK FROM *^*^"" aRSSNWICH *'*-AU» GMimrWTCH H. M. & H. M. a. Albany +4 54 59.99 Denfer +6 59 47.63 Algiers -0 12 08.55 Edinburgh +0 12 44.2 Allegheny, Penn .. + 5 20 02.93 Glasgow +0 17 10.55 Ann Arbor, Mich. + 5 34 55.19 Madison, Wis +5 67 37.93 Berkeley, Calif. . . + 8 09 02.72 Madrid + 14 45.12 Berlin -0 53 34.85 Mexico +6 36 26.73 Bombay -4 51 15.74 New York +4 55 53.64 Cambridge (Eng.) - 00 22.75 Paris - 09 20.97 Cambridge (Mass.) + 4 44 31.05 Philadelphia +5 00 38.51 Cape of Good Hope -1 13 54.76 St. Petersburg ...- 2 01 13.46 Chicago +5 50 26.84 Washington +5 08 15.78 16. Give from the table the difference of local times of Green- wich, England, and of each of the following places and state whether the time is earlier or later than Greenwich time : Albany ; Algiers; Ann Arbor; Berlin; Bombay; Chicago; New York; Paris; St. Petersburg. 17. What is the longitude in degrees (°), minutes and sec- onds (") of arc of Albany? of Algiers? of Cambridge (Mass.)? of Washington? 18. Which of each of these pairs of places has the earlier local time and how much earlier is this time : Albany and Allegheny? Chicago and Paris? Ann Arbor and Berkeley? Edinburgh and Madison? Ann Arbor and Bombay? Cape of Good Hope and Bombay? Chicago and Cambridge,Mass.? Paris and Bombay? 19. Give the differences of longitude in each case of problem 18. 20. Solve other similar problems on the table. LONGITUDE AND TIME 303 §183. Standard Time. For convenience of railway traflSc a uniform system of time- keeping, known as Standard Time, was agreed upon in 1883 by the principal railroad companies of North America. It was decided that places within a belt of 15° extending (roughly) 7^° on each side of the 75th meridian west of Greenwich should use the time of the 75th meridian. All places in similar belts extending about 7^° on each side of the 90th, of the 105th, and of the 120th meridian should take the times of those meridians respectively, and hence should have times just 1 hr., 2 hr., and 3 hr. earlier (less) than the time of the 75th meridian time belt. This system has now been generally adopted by most civilized coun- tries. In practice, however, the dividing lines of the time belts are uregular lines running through railroad terminals. The fol- lowing map shows the time belts and the names used to dis- tinguish the times of the several belts which cover continental United States. In this system the times at all places in the United States differ only by whole hours. Pacific Time 120° MouifTAiN Time 105° CsNTBAii Time Eastern Time 75° FlOUBB 192 1. When it is 8 o'clock a.m. (Standard Time) at Chicago, what is the time at each of the following places : New York? Pittsburg? 304 RATIONAL GRAMMAR SCHOOL ARITHMETIC Sb. Louis? Omaha? Denver? Spokane? San Francisco? (See the map, Fig. 192.) . 2. When it is 9 hr. 10 min. 45 sec. p.m. at Omaha, what is the time at New Orleans? Philadelphia? Buffalo? Washington? Austin (Tex.)? Boise City? Los Angeles? El Paso? Salt Lake City? (Refer to your Geography.) 3. Answer the questions of problem 2 for 2 : 15 p.m. at St. Paul ; for 3: 25 a.m. at Dodge City, Kan. 4. Answer for other places marked on the map such questions as are asked in problems 2 and 3 for Omaha, for St. Paul, and for Dodge City. The circle SqWr (Fig. 193) represents the earth and the curved lines represent the hourly meridians running from the equator, qr^ and converging toward the poles S and iV. If represents some place in the northern hemi- sphere, SqJVr is the meridian of the place and BQPR may be im- agined to represent the meridian of the sky {hour circle of the sun) , on which the sun, T, is situated. The hour circle of the sun in the apparatus is held in place at ^. If now the crank F is turned so as to carry q forward and downward FIGURE 193 through E to r, the sun's meridian standing stationary, the motions which cause the differences of time and the changes of day and night may be understood, by recalling that only the half of the globe which is turned to the sun is light (in day). /S 6^ iV denotes the prime^meridimi. 5. As the globe is standing in Fig. 193, what time is it at O? at places on the 105th meridian? on the 60th meridian? on the 15th meridian east* of Greenwich? on the 45th east? 6. If the meridian SGN is continued around on the other side of the globe, what will its number be? 7. Remembering that any place on the earth, as O, turns com- pletely round through 360° in 24 hr., how long does it require * Remember that east is the direction toward which the globe is turning, that is east- ward means from E toward through r and around to E again. LONGITUDE AKD TIME 305 the Space between any two adjacent meridians shown in Fig. 193 to pass under the sun's hour circle? 8. If a man should start from some place on the prime meridian (say London) on Friday noon and move westward just as fast as the globe turns eastward, what time (by the sun) would it be to him during his journey all the way round the globe? What hour and day would a Londoner call it when the traveler returned 24 hr. later? The problem raises the question, ''Where should the traveler have changed his date so that his date might agree with that of his starting place when he returns?" The answer is, "It has been agreed that the date should change at the 180th meridian. " When vessels cross this meridian from the east toward the west they add a day to their reckoning. If they cross at noon on Friday, Friday noon instantly becomes Saturday noon. . Crossing from the west toward the east, they repeat a day. In the case mentioned, Friday would ''be done over again." The 180th meridian is for this reason called the Date Line. Trace it round the earth in your Geography. 9. The date begins on the 180th meridian of longitude and travels westward around the earth. Supposing the day begins at midnight, give the date and hour on the 180th meridian, when it is Wednesday, Oct. 15, 10 P. M., standard time, in Chicago; in Washington; in Denver; in San Francisco. 10. The U. S. Supreme Court has decided that legal time at any place is the local time of the place. A man in Akron, 0., insured his house at 11:30 A. M. "standard" (90th meridian) time and the house took fire at that moment and was destroyed. The policy stated that the insurance would be in force at and after 12 o'clock (noon) of the day of insuring. The difference between Akron local time and 90th meridian time is 33 minutes. Would the law require the insurance company to pay thv^ policy? Give reason for your answer. 11. If every place on the earth has the time of that meridian of the map, p. 296, nearest to it, what will be the time of the fol- lowing places when it is 3:25 P. M. in England? (1) Washington? (2) Boston? (3) New Orleans? (4) Denver? (5) Rio Janeiro? ((3) Cuba? (7) Japan? (8) Philippine Islands? 306 RATIONAL GRAMMAR SCHOOL ARITHMETIC MENSTTBATION §184. Areas, Roofing and Brickwork. 1. Give the rule for computing the area of a square from its measured sides; a rectangle; a parallelogram; a triangle. (See pp. 102-104 and 179.) Fig. 194 represents a plot of the streets and blocks of a part of a certain city. The streets are all 75 ft. wide between sidewalks. The numbers written on the lines indicate their lengths in feet. The dotted lines show how to divide up the areas into parts for computation. Figure 194 2. If $240 per foot of frontage on both 3rd and 4th streets was paid for block B (Fig. 194), how much did the block cost per square foot? per square yard? 3. Find the area in square yards of other parallelograms, such as 0, P, etc., of Fig. 194. 4. Find the area in square feet of the following triangles of Fig. 194: E; G; I; J; K; L; abc. Other areas of Fig. 194 require a knowledge of trapezoids. 1 Definitions. — A four-sided figure (a quadrilateral) like Fig. 195. having one pair of parallel sides, is called a trapezoid. The altitude of a trapezoid is the distance square across Figure i95 between the two parallel sides (called the bases.) MENSURATION 307 5. Study tlie trapezoids of Fig. 196, and find how to get the length of EF^ the line connecting the middle points of the two non-parallel sides, from the lengths of the two bases. After com- puting EF^ find for each trapezoid a rectangle whose area equals the area of the trapezoid. Find the areas of the trapezoids of Fig. 196. (B^^^t^/K Figure 196 Definition. — The lengths a, 6, and c of the last trapezoid are called its dimensions. Note. — J the sum of x and y is written \{x-\-y), n times the half sum is written \n{x + y). 6. How do the altitude and the sum of the bases of a trapezoid compare with the altitude and base of a parallelogram having an ^•7 V ^"""'7 V V — ^ area equal to the area of the 5^-- \ 5^:: ^ ^--^^ trapezoid (Fig. 197)? . Figure 197 7. Supposing a, ^, and c are the dimensions, in feet, of the trapezoids of Fig. 197, what are the areas of these trapezoids? 8. Find the areas in square feet of C\ of 7>; of J/; of N\ of H (Fig. 194). 9. Calling Z the area of any trapezoid, whose bases are h and c and whose altitude is a, write an equation showing how to find Z from «, by and c, 10. Find the area in square yards of ^, ^, N, S, T, Fig. 194. Definition. — A square of roofing means a 10' square of roof surface, or 100 sq. ft. A shingle is said to be laid 4", 4 J", or S'' to the weather when the lower end of each course of shingles on the roof extends 4", 4^", or b" below the course next above it. 11. Draw two perpendicular center lines, and with the aid of the dimen- sions given in the left part of Fig. 198, complete an enlarged drawing to a con- venient scale, of the development of the roof (shown on the right), figurbiss 308 RATIONAL GRAMMAR SCHOOL ARITHMETIC 12. If 1000 shingles laid 4" to the weather cover a square of roofing, how many shingles will be needed to cover a square, if laid 5" to the weather? ^" to the weather? 3" to the weather? 3r? 13. The dimensions on the development (Fig. 198) being in feet, find the cost of the shingles, at $1.10 a bunch of 250, needed to cover the four sides of the deck roof (Fig. 198). Find the cost of enough tin to cover the 12' x 16' flat deck at 15^ a 20"x28" sheet, the long sides of the sheets being laid parallel to the long side of the deck, and allowing 10% loss for joints and overlap at edges. 14. When shingles are laid 4" to the weather, 1000 shingles are estimated to cover a square (of roofing). Find the number of shingles, laid 4'' to the weather, needed to cover a roof which, with gables, is made up of the following parts: Shapes Altitudes Bases 2 rectangles 10' 40' 2 psvraUelograms 12 ' 26 i ' 2 parallelograms 12' 14' 2 triangles 7' 14' Shapes Altitudes Bases 4 triangles 8i' 17' 2 trapezoids 12 12' and 20)' 2 trapezoids 12' 5' and 13i' 8 trapezoids 12' 6* and 14i' 90c a bunch 15. Find the cost of the shingles of Prob. 14 of 250. 16. Ten years after building this house a new roof was put on it. It cost $1.25 per M. to remove the old shingles, $3 per M. to dip the new ones, and $2.76 per M. for the labor of putting on the shingles. How much did it cost to remove tlie old shingles and to re-roof with dipped shingles? 17. If it takes 14 bricks per square foot of outside surface to lay a 2-brick wall, how many bricks will be needed to lay the side wall of the build- ing shown in Fig. 199, deducting for 5 windows each 3^' x 6J' and for 5 windows each 3 J' x OJ'. 18. Make other similar problems from your own measurements or from dimensions obtained from an architect or builder. figure 199 MENSURATION 309 19. The walls of a square brick house extend 6' into the ground and rise 24' above the ground to the eaves. From the bottom to 12' above the ground the walls are 3 bricks thick and from this line to the eaves they are 2 bricks thick. Find the cost of laying the bricks of these walls @ $4.25 per M. Note. — No allowance for openings of any kind is made in figuring the cost of laying bricks in such walls, and outside dimensions are used, thus counting corners twice. 20. From the number of bricks found in problem 21 deduct for the following openings : Downstairs Upstairs 1 door 3' 6" X 6' 9" ; 2 doors 8' 6" X 6' 6" : 1 door 3' X 6' 6" ; 1 window 5' X 6' 6" ; 2 windows 5' X 6' 6" ; 12 windows 3' X 6' ; 9 windows 3' X 6' ; and 1400 bricks for double counting of comers, and find the cost of the bricks @ $12^ per M. 21. My friend's house has two octagonal (8-sided) towers, the roofs being made up of equal triangles (Fig. 200). Find the number of slates for both roofs, 240 to the square (100 sq. ft.), and cost @ $2.75 a hundred. 22. Draw the develop- ment of the roof and tower to a scale J" to 1', as shown in the figure. 23. From the eaves to the ground the towers are octagonal prisms, built of brick, all faces being rect- angles 6' X 40', and each prism has fnur faces exposed to the weather. How many square feet of brick surface are in the 8 exposed faces, not allow- ing for windows? Find the cost of laying the brick @ $6.25 per M. 24. Deduct for 16 windows, each 3' 6" x 5' 8", and for 4 win- dows V 6" X 3', and find the number of bricks in the two layers, and their cost @ $14 per M. 3BB BBS BBBB BBBB Figure 200 310 RATIONAL GRAMMAR SCHOOL ARITHMETIC §186. Land Measure. - Review §78, pp. 119 and 120. The law requires land to be marked out or surveyed in divisions of the form of squares and rectangles. In the western states the land has been surveyed in accordance with this law. To mark out the largest squares, north and south lines, called meridians^ are first run 24 mi. apart and marked with comer- stones, or by trees, or other permanent objects. East and west lines, called base lines, are then run at right angles to these meridians at distances 24 mi. apart. This would divide the land up into 24-mi. squares were it not for the convergence of the meridians toward the poles of the earth. Notice this on a map in your Geography and on the map of Fig. 192. Each 24-mi. tract is then divided into 16 nearly equal squares, called townships, by running north and south, and east and west lines through the quarter points of the sides of the large tract. How long is a township? how wide? how many square miles does it contain? Certain meridians, called principal meridians^ are run with great care, and these principal meridians govern the surveys of lands lying along them for considerable distances both toward the east and toward the west. The tiers, or rows, of townships running north and south along the principal meridians are called ranges. The first tier on the east is called range No. 1 east, and is written KIE; the second range is No. 2 east, written E2E, and so on. Point out on the drawing. Fig. 201, EIW; E2W; R3W; E2E; E4W. Certain base lines are run with great care and are called standard base lines. The rows of townships running east and west are numbered with refer- ence to these standard base lines. A township in the first row north of a base line is town- ship No. 1 north, and is written TIN; one in the second row south is called township No. 2 south, written T2S, and so on. Interpret the following symbols and point out on the drawing, Fig. 201, the townships indicated: T3N; T4N; TlS; T2N. a: Standi Ul Ul rd Line 4 4 4 4 4 4 c c 3 3 3 •1 3 3 2 2 2 2 2 1 1 1 -- Ba se Lin e f I 1 Fig UBE 9 1 1 J MENSURATION 311 A township is identified by giving its number and range from some standard base and principal meridian. Point out these townships on the drawing, Fig. 201: TIN, R2W; T3N, RlW; T4N, R2E; TlS, R3W. The law also requires townships to be sub- divided into smaller squares, called sections. Sections are numbered, beginning at the north- east corner and run- ning toward the west to 6, then 7 is just south of 6, and so on, as in Fig. 202. Sections are then sub- divided into quarters (see section 16), half-quarters (see sections 13 and 26), and quarter-quarters (see section 29). 1. Referring to Fig. 202, read and write the descriptions of the divi- sions of section 16; of section 26; of 13; of 29. Whatever deviations there may be from exactly 640 A., in the sections of any township, due to convergence of meridians or other causes, are required by law to be added to, or sub- tracted from, the north and west rows of half-sections. These tracts are then not called fractional sections, but are called Zo^5, and are numbered in regular order as shown in Fig. 202. A section then always means exactly 640 A. Any fractional part of a section means the ' corresponding fractional part of 640 A. The lot, on the contrary, must always be measured FiGuaa 203 before its area is known. Lot 16 Lot 15 321 A Lot I4t iLot 13 Lot 12 ^U' ^% 6 5 4 3 2 1 7 8 9 10 II 12 S5 >s 18 17 15 14 h I p i 19 20 21 22 23 24 30 28 27 25 -I n ^ ft t V ^ 3i 32 33 34 1 35 36 FIGURB 202 L48 38 J.CH L47 41 URCH L49 4-IA J. L50 82 A 40 80 Q^g.M. EVANS LSI 8IA *«|S.PBRRY 8iA H.BE JbWQSh L 46 79 A W. BROWN 80 80m 160 Seel ion Line ^ ^^l^ 160 80 VENS C.PAfKS 160 D Line L45 79 A J. BRADEN 60 g40 H. BRADEW . L44 39A CI 160 t 160 E80 isO 160 80 > 580 L43 38A 80 o -C80 312 RATIONAL GRAMMAR SCHOOL ARITHMETIC 2. Following is an assessment list of farm property. Fill the blanks from Fig. 203: OWNBB DESCBIPTION NO. ACBBS H. Peabody EJ SEJ Sec. 8 H. Peabody E J NEJ Sec. 8 J. White EJ SEJ Sec. 5 J. James SE^ NE^ Sec. 5 J. James Lot 43 .... 0. Gibson '. Lot 44 .... 0. Gibson SWi NE^ Sec. 5 .... 3. Make out an assessment list for all owners in Sec. 6. 4. Make and solve a similar problem for owners in Sec. 7. 5. Correct the mistakes in this erroneous assessment list: OWNKB DESCBIPTION NO. ACBI8 S. Perry WJ NWJ Sec. 7 81 J. Hay Wi SWi Sec. 6 88 J. Hay Wi SWi Sec. 7 80 H. Ochiltree SWJ NW^ Sec. 6 41 H. Ochiltree SEJ^ NE J Sec. 6 40 H. Ochiltree EJ SE} Sec. 6 80 J. Church NWi NWi Sec. 6 38 J. Church NEJ NWJ Sec. 6 41 §186. Volumes. Definition. — The volume of any figure is the number of cubical units within its bounding surfaces. 1. Give the rule for finding the volume of a square prism (called also a rectangular parallelepiped) . (See pp. 112, 113.) 2. Give a rule for finding the volume of an oblique parallele- piped (Fig. 148, p. 277) having the same base and the same altitude as a given rectangular parallelepiped. (See Figs. 150 and 151, pp. 277 and 278.) 3. How would you find the volume of a hollow beam 12 ft. long, halving a cross section like (1) Fig. 05, p. 132? (2)? (3)? (5)? (6)? (4)? (9)? MENBURATION 313 Scale 1 :6 4. Giv-e the volu-mes of the beams of problem 3, if the length of the beam in each case is I feet. 5. A model of a square prism (like Fig. 145) having a base 2 in. square and an altitude of 8 in. will contain how many cubic inches of sand? 6. The model of a right circular cylinder, made as shown in the scale drawing of Fig. 204 and pasted along the flap, DF, and around one end with a strip of paper, was filled with sand and poured into the empty model of the square prism of problem 5. It filled the square prism a little more than f full. About how many cubic figubb ao4 inches are there in the model of the cylinder? 7. It is shown in geometry that the volume of any right cir- cular cylinder is. 7854 (= j) times the volume of a square prism of the same altitude and having for one side of its base the diameter of the cylinder. Find the volumes of these circular cylinders : / £ (1) Diameter 4", altitude 6" (4) Diameter ISf, altitude 8" (2) " 7" " 7" (o) " 6' " 10.5' (3) " 10" '♦ Sf (6) " 2r " a 8.. How long is AB? How many square inches in the rectangle ADFG (Fig. 204)? How many square inches are there in the en- tire outside surface of the cylinder? 9. The inside diameter of the cylindrical water tank of a street sprinkler is 3.6 ft., and its length is 10 ft. ; how many liquid gallons does the tank hold? 10. The tank of the sprinkler is filled through a hose of 2^' inside diameter from a hydrant from which the water flows at the rate of 300 linear feet per minute. How long will it take to fill the tank? Suggestion. — In 1 min. a cylinder of water as large as the inside of the hose and 300' long flows into the tank. 11. Find the weight of an iron rod ^" in diameter and 24' long, if iron weighs 450 lb. per cubic foot? 314 RATIONAL GRAMMAR SCHOOL ARITHMETIC 12. How many cubic inches of air are there in the hollow tire of a bicycle wheel 28" in diameter (from center line to center line of tubes), the hollow having a diameter of If"? 13. How many cubic inches of rubber are there in the hollow tire of a 32" automobile wheel if the inside diameter of the cylin- drical tube is 3" and the outside diameter is 4 inches? ^ ^ 14. Fig. 205 is a scale drawing of a pattern "IT for the paper model of a cone whose base is to ^^ be a circle 2" in diameter and whose sloping ^ side from the apex to the base is to be 8". The \ arc AHB is just as long as the circumference \ of the base. Compute the length of the cir- ^^^^^^^3^ cumference of a circle whose radius is 2", and Scale i^/Si^ ^^ another whose radius is 8", and find the vJ/ ratio of the first to the second. Figure 205 15^ What part of the whole circumference (with center C) is the arc AHBl What part of 360'' is the angle AGBl 16. Make a right angle, as HOX^ and bisect it (See Problem II, p. 176). CD is the bisector. Bisect angle HCD and obtain CB. 17. With C as center and with a convenient radius, as CE^ draw the indefinite arc PNME. Put the pin-foot on G and mark an arc across arc OP at N. Draw CiV and prolong it. This makes angle GCN equal to angle GCM. Now with (7 as a center and with 8" as a radius, draw the arc AHB. Prolong CHy make HO = 1", and draw the lower circle. Provide the flap and paste up the model of the cone. If such a model is carefully made and the height is measured and if the model of a cylinder has the same height and the same base, the model of the cone will be found to hold ^ as much sand as does the model of the cylinder. It is proved in geometry that the volume of any cone equals ^ of the volume of a cylinder having an equal base and an equal altitude. 18. Find the volumes of these circular cones: (1) Diameter of base 3", altitude, T' (2) " " 6" " 20" (3) " '' 6.8" " 2.25' (4) '' '' 3.95" " 10.25" MENSURATION 315 19. How many cubic inches of water are there in a funnel- shaped vdssel, if the water is 4" deep and the diameter of the surface of the water is 4.5"? (See Fig. 206). 20. How many cubic inches of water will it take to fill the same vessel to twice the depth, or 8"? (See Fig. 206.) 21. How many cubic inches had to be poured into the vessel to fill it to 8" depth if the depth of the water was 4" at the beginning? figubk aoe 22. A mountain peak has the shape of a circular cone whose altitude is 1.25 mi. and the diameter of whose base is 2.35 miles. Find its volume in cubic miles. 23. How much water will it take to fill a conical vase 10" deep and 3.225" across at the top. 24. The conical tower of a building is 24.75' across at the base and 36.8' high. How many cubic feet of space does it occupy? 25. If a cord or waxed tape (bicycle repair tape) be wrapped around the curved outside surface of a half croquet ball, as a top is wound, until the surface is covered, and then if the same cord, or tape, be wrapped around on the flat circular base beginning at the center until the circle is covered, the length of the former cord will be found to be just twice the latter. The area of the surface of the wliole ball is then how many times the area of the circular section of the ball? What radius has this circular section? It is proved in geometry that the area of the surface of a sphere equals 4 times the area of a circular section going through the center of the sphere. 26. The radius of a ball is 1^"; what is the area of the greatest circular section of the ball? What is the area of the surface of the ball? 27. Measure the circumference of a baseball and find how many square inches of leather there are in the cover of the ball. 28. How many square inches of paint will it take to cover the surface of a globe of 10" radius? of 10" diameter? 29. The average diameter of the earth is 7918 mi. ; how many square miles are there in its surface? 316 BATIOlSrAL GRAMMAR SCHOOL ARITHMETIC 30. Calling s the surface and r the radius of a sphere, write an equation showing the relation between s and r. It is seen on p. 277 that the volume of a triangular right prism equals i the volume of a square prism whose base and alti- tude are equal to the base and the altitude of the square prism. 31. Find the volume of a right triangular prism whose altitude is 18" and whose base is a triangle having a base of 8" and an altitude of 5" inches. FIGUBE207 PIOUBE808 j, ^ triaugular pyramid be care- fully modeled (Fig. 207) and filled with sand it will be found that just 3 times the volume of the pyramid is equal to the volume of the model of a triangular prism (see Fig. 208) of equal base and equal altitude. It is proved in geometry that the volume of any pyramid equals i of the volume of a prism having an equal base and an equal altitude. Notice in Fig. 208 how a triangular prism may be completed on a triangular pyramid having the same base and the same altitude as the prism. 32. Calling V the volume, B the area of the base, and a the altitude of a triangular pyramid, write an equation, showing the way V would be computed from B and a. 33. Find the volume of a pyramid whose base contains 16 sq. in. and whose altitude is 12 inches. 34. The Great Pyramid of Egypt is 481 ft. high and its base is a 756' square. If it were solid and had smooth faces, how many cubic feet of masonry would it contain? 35. At the close of the nineteenth century the United States had 195,887 mi. of railroad. How many times would these rail- roads, if placed end to end, encircle the earth? (Use v = 34^, and the radius of the earth = 3959 miles). 36. The ties used for these roads would contain wood enough to make a pyramid of the same shape 1395' high with a 2192' square for its base. How many cubic feet of wood were used for the railroad ties? 37. It has been computed that the materials used for the road beds for these railroads would make a solid pyramid 2470 ft. high COKSTRUCTIVE GEOMETRY 317 How many cubic feet and having a 3870 ft. square for its base, would this make? If the entire surface of a globe, or sphere, were divided up into small triangles like the one shown in Fig. 209, and the sphere were cut up by planes cutting along the curved sides of the triangles and passing through the center, 0, the volume of the sphere would be divided up approx- imately into small triangular pyramids, having their vertices at the center. The volume of each pyramid would be the product of its triangular base by J of the figure 209 the radius of the sphere. The sum of the areas of all triangular pyramids would equal the surface of the whole sphere and the sum of all the volumes of the pyramids would equal the sur- face of tlie whole sphere multiplied by i of the radius of the sphere. 38. Calling V the volume and r the radius of a sphere, give the meaning of the formula : o o 39. Find the number of cubic inches in a sphere of 2" radius. 40. How many cubic inches in a croquet ball 4" in diameter? in a tennis ball 1.75" in diameter? in a baseball 2.1" in diameter? in a globe 10.15" in diameter? 41. Calling t^e sun, the moon and the planets all spheres with diameters in miles as rn the following table, compute their cir- cumferences in miles, their surfaces in square miles and their volumes in cubic miles : Moon 2,160; Mercury 3,030; Venus 7,700; Earth 7,918; Mars 4,230 Jupiter 86,500 Saturn 73,000 Uranus 31.900; Neptune 34,800; Sun 866,400, §187. Constructive Geometry. Problem I. — To find the center of a. given arc. Explanation. — Let AB^ Fig. 210, be the given arc whose center is to be found. Mark any three points, as C, E, and D, on the arc. Draw the straight lines CE and ED. Bisect each of these lines as in Problem VI, pp. 100 and 101, and prolong these bisectors until they intersect as at O. O is the required center. Definition. — The lines C-EJand ED, each of which connects two points of the arc, are called chords of. FiGUBB 21U the arc. 318 EATIOXAL GBAMMAR SCHOOL ARITHMETIC To solve this problem is it necessary actually to draw the chords? How could you find the center of a circle that would go through any three points not in a straight line? Mark 3 points not all in the same straight line and draw a circle through them. Problem II. — To bisect a given arc. Explanation.— Let AB, Fig. 211, be the given arc. With ^ as a center and then with B as a center and with a radius greater than the distance from A to the middle of the arc AB^ draw the dotted arcs as indicated. Lay a ruler on the intersections of the two arcs and draw a short line across the arc, as at C. C is the mid-point of the arc, and arc -4C = arc CB. X v FIGUBB 211 Problem III.- triangle. -To circumscribe a circle around an equilateral Explanation.— Draw an equilateral triangle as in Problem VII, p. 102, and bisect two of its angles as shown in Fig. 212. With the intersec- tion of the bisectors as a center and with a radius equal to the distance from this intersection to any vertex, draw a circle. This circle is said to be circumscribed around the triangle. Problem IV. — To draw a trefoil. Figure 212 Figure 213 Explanation.— Draw an equilateral tri- angle and bisect one of its sides, as shown in Fig. 213. With the upper vertex as center and with a radius equal to the distance from this vertex to the middle of the bisected side draw an arc of a circle around until it touches the sides of the triangle both ways. Draw arcs around the other vertices in the same way. If desired, other circles, with slightly longer radii, may be drawn just outside of these until the arcs come together but do not cross. Problem V. — To construct a square and to draw a quatrefoil (a four-foil) upon it. Explanation.— Prolong BA through A far enough to draw a perpendicular, as AC, to AB at A. Draw this perpendicular and make A(J =^ AB With AB as a radius and (1) with C as a center, then (2) with ^ as a center, draw two intersecting arcs at D. Draw CD and BD and complete the drawing as shown in Fig. 214. Figure 2U CONSTRUCTIVE GEOMETRY 319 Problem VI. — To draw the designs of Fig. 215. Explanation. — Draw a square like the dotted squares'of Fig. 215. Study the two draw- ings, decide where the centers of the arcs are and complete the designs. Figure 215 Problem VII. — To draw a sixfoil. Explanation.— Draw a circle, like the dotted one in Fig. 216, with a radius as long as one side of the regular hexagon is to be. Draw the hexa- gon (see Problem XI, p. 104). Bisect a side of the- hexagon, and, using the vertices of the hexagon as centers, complete the sixfoil, as shown. Outside arcs may be added if desired. FlGUBB 216 Problem VIII. — To construct a right triangle. The symbol "perpendicular ular to." 1 means to," or 'perpendicular," "is perpendic- CoNSTRUCTioN. — Draw the line CDlAB, Fig. 217. Then draw the line OF connecting any point, as G, of the line CD with any point of AB, as F. Definitions. — The longest side, that is, tlie side opposite the right angle of a right triangle, is called the hypothenuse. Figure 217 What side of the triangle, GOF, is the hypothenuse? What angle is the right angle? FXOUBB 218 Problem IX. — To construct a right triangle having a given hypothenuse. Construction. — Let the line AB, Fig. 218. 5 denote the given hypothenuse. Bisect AB as at O and with O as a center and OA as a radius, draw a semicircle. Connect any point of the semi-circumference, as E, D, or C, with A and with B. Any such triangle is a right triangle and the line AB is its hypothenuse. 320 RATIONAL GRAMMAR SCHOOL ARITHMETIC 218. Point out the right angle of each of the three triangles of Fig. Figure 219 Problem X. — To construct a right triangle, having given the two sides which include the right angle. A « Construction.— Let the two given sides '^—^ be a and b, Fig. 219. Draw BC 1 DE, and make OA==a and OB = b. Connect A with B, BOA is the re- quired triangle. How may an isosceles right triangle be constructed? Problem XI.— To find the relation of the squares of the sides of an isosceles right triangle. Construction.— Construct an isosceles right trian gle and on each of its three sides draw a square. Draw the dotted lines and cut the side squares as shown in Fig. 220. Fit the pieces over the large square on the hypothenuse. If the area of each side square were 9 sq. in. what would be the area of the large square? FIGURB 220 Problem XII. —To find the relation of the squares of the three sides of any right triangle. F16URB 221 Construction. — Construct any right triangle and then construct a square on each of its three sides. Cut the side squares as shown.* Fit the pieces over the s^quare drawn on the hypothenuse, as indicated by the dotted lines in Fig. 221. What single square has an area that equals the sum of the areas of the squares on the two shorter sides of any right triangle? ^he center of a square Is the lutersectiou of Us diagonals. SQUARES AND SQUARE ROOTS 321 1. Denoting the length of either short side of Fig. 220 by a, what denotes the area of the square drawn upon this side? 2. Denoting the length of the hypothenuse of Fig 220 by /*, what denotes the area of the square drawn on the hypothenuse? 3. Write an equation from Fig. 220, showing the relation between a* and A*. 4. Denote the lengths of the three sides of any right triangle (Fig. 221) by a, 5, and h (A being the hypothenuse). What will denote the areas of each of the squares on the three sides? 5. Write an equation showing the relation of the squares of the sides of any right triangle. PROBLEMS 1. The sides of a right triangle are 3" and 4"; what is the length of the hypothenuse? SUOOESTION.— 7i2 = 3* + 42 = 9 -f 16 = 25 ; what is the value of h? 2. The hypothenuse of a right triangle is 10", and one of the sides is 6" ; what is the other side? SUOGESTION.— lO^ = a2 + 6^ ; find the value of a? 3. The sides of a right triangle are denoted by a, J, and the hypothenuse by h; find the unknown side in each of the following right triangles: (1) a = 9, 5 = 12; (4) a = 32, 5 = 24; (2) a = 12, h = 20; (5) b = 21, A = 35; (3) b = 15, h = 25; (C) b = 27, h = 45. Before the hypothenuse of a right triangle, whose sides are 34 and 26, can be computed, it is necessary to know how to find the square roots of given numbers. §188. Squares and Square Boots. DKFiNmoN. — The product obtained by using any number twice as a factor is called the sqvure of that number Thus, 36 is the square of 6, because 6, used twice as a factor gives 36 (6x6 = 36). The square of a number as 6 is often written thus, 6^. What does the small 2 show? 322 RATIONAL GRAMMAE SCHOOL ARITHMETIC The following squares should be committed to memory : Squares of Units Squares of Tens Squares of Hundreds 1^= 1; 10* = 100; 100*= 10000; 2«- 4; 20*= 400; 200*= 40000; 3«= 9; 30* = 900; 300*= 90000; 4* = IG; 40» « 1600; 400* = 160000; 5««25; 50« = 2500; 500* = 250000; 6» = 36; 60« = 3600; 600* = 360000; 7« = 49; 70« = 4900; 700« = 400000 ; 8« = 64; 80« = 6400; 800* = 640000; 9« = 81; 90* = 8100; 900* = 810000. 1. Write all the 5 pairs of number^ which, multiplied together, give the product 36; the product 16; the product 64; the product 49. Note.— Write the pairs of factors of 86 thus : 1 and 36 ; 2 and 18 ; 3 and 12; 4 and 9; 6 and 6. Proceed similarly with the rest. Definition.— The square root of a number is one of the tico equal factors of it. The sign of square root is -s/, called the radical sign. Thus, y/'Z^ means the square root of 25, which is 5. 2. Give the square roots of the following numbers: 9; 16; 49; 64; 81 ; 400; 2500; 3600; 160000; 490000; 810000. To find the square root of a number not in the table above, it is necessary first to learn how the square of a number is formed from the number. From the table answer the following questions: 3. How many digits are there in the square of any number of units? of tens? of hundreds? 4. Find the square of each of the following decimals: .1, .3, .5, .7, .9, .01, .03, .04, .06, .07, .09, .001, .003, .005, .006, .007, .009. 5. How many decimal places are there in the square of any number of tenths? of hundredths? of thousandths? 6. If, then, the square of any number of units contains only units and tens, what places of any number that is a square must contain the square of the units of its square root? 7. What places must contain the square of the tens? of the hundreds? of the tenths? of the hundredths? of the thousandths? SQUARES AND SQUARE ROOTS 323 8. If, then, we separate a number, whose square root is desired, into two-digit groups, beginning at the decimal point and proceed- ing both toward the left and toward the right, what one of these groups must contain the square of the units of the square root? the square of the tens? of the hnndreds? of the tenths? of the hundredths? 9 Find the square of 46. Meaning of Common Method 46 = 40 + 6; 462 = (40 + 6)2 =(40+6) (40+6) = 40X40 + 40X6+6X40 + 6X6 = 402 _|_2 X (40 X 6) + 62 = 1600 + 480+36 = 2116. Show the meaning of the parts of this sum in Fig. 232. Thus it is seen that the square of a two-figure number is the sum of (1) the square of the tens, (2) twice the product of the tens by the units, and (3) the square of the units. This shows that if any two-figure number be denoted by f + w, where t denotes the tens and u the units, the square is formed thus : t + u ' ^ t + u 2116 = 2116. FlGUBl^ 222 40+6 40+6 240 + 36 1600 + 240 1600+480 + 36 Show the meaning of the equation Fig. 223. tu + u^ t^ + tu by Figure 223 10. Find the square root of 2116. Convenient Form 46 = required root ; 2116 denoted by t^ + 2tu + u^ ; 1600 = greatest square (of the table) in 2116; 2f = 80 516 contains 2tu + u^, where ^ = 40 ; 2^ + u = 86 516 denoted by 2tu + u^ , wliere m = 6 ; Check: 46 X 46 = 2116. Shortened Form 21 '16 I 46 = required root 80 16 CJieck: 46x46 = 2116. 86 516 516 324 RATIONAL GRAMMAR SCHOOL ARITHMETIC 11 Find the square roots of the following numbers: (1)484; (3)1156; (5)1296; (7)7569; (9)9604; (2) 729; (4)1225; (G) 5G25; (8) 9409; (10) 110224. 12. Find the square root of 2079. 3G. 80 5 Hn uo. ^.6 90.6 20'79.86' 16'00 45.6 4'79. 4'25. 54.36 5436 Ans. -^2079.36 = 45.6 Check: 45.6 X 45.6 = 2079.: CONVENIEKT FORM Explanation. — Separate the number into two-digit groups. Beginning on tbe extreme left, subtract the greatest square of tens in 2iO hundreds, viz., 16 hundreds (=40^), and write 4 tens on the right as the first root digit. Double the 4 tens and use the result 80 as a trial divisor. 80 is contained in the remainder, 479, 5 times. Write 5 as the 2d root digit and also add it to the 80. giving 85 as the complete divisor. Double the part of the root found, 45, giving 90 for the next trial divisor and complete the steps as before. 13. Find the square root of 1578 to 3 decimal places. Convenient Foem 15'78.00'00'00M39.724 + 60 9 678. 66 78.0 .7 621. 1 57.00 Explanation.— Annex zeros and proceed as before. s:7~L 7»v40 .02 79.4a 55.09 1.9100 Check: 39.724 X 39.724=1577.996176. rem. = .003824. 1.5884 1578. 79.440 .321600 79.444 .317776 .003824 remainder. In actual practice the decimal point is needed only in the root 14. Find the square roots of the following numbers: (I)- 1900.96; (2) 4719.G9; (3) 5055.21; (4) 61.1524; (5) 75.8641; (G) 79.9236; (7) .5476; (8) .458329; (9) 1.216609.* SQUARES AND SQUARE ROOTS 325 15. Find to 3 decimal places the square roots of the following numbers : (1) 5; (4) .85; (7) 1683; (10) 1.85; (2) 7; (5) .25; (8) 6875; (11) 26.79; (3) 15; (6) 125; (9) 7328; (12) 64.893 16. Find by multiplication the values of the following expres- sions : (1) (if; (3) (*)'; (5) (^)'; . (7) {±r; (2) (f)^ (4) (if)'; (6) (H)'; (8) (^)'. 17. Make a rule for finding the square root of a common fraction. 18. Find, without reducing the common fractions to decimals, the values of the following expressions and prove them by multi- plication: (1) ^f; (4) ^TI|; (7) v/^v;; (2) y/h (^) v/^5 (8) ^/4H> (3) v/T|; (6) ^^; (9) p 19. The sides of a right triangle are respectively 34" and 26" long, how long is the hypothenuse? 20. The center pole of a circus tent is 35' high, and a guy rope is stretched from the top of the pole to a stake 56' from the bottom. How long is the rope, supposing the ground level and the rope straight, allowing 4' for tying? 21. 50' of the top of a tree standing on level ground is broken by the wind and remains fastened to the stump. If the top strikes the ground 30' from the stump, how high was the tree? 22. A horse is staked out by a rope 40' long to the top of a stake 15" high. Over what area can the horse graze? 23. The vertical mast of a hoisting derrick 35' high is held in position by four guy ropes staked to the ground at the four corner points of a square. The stakes are 75' from the bottom of the mast. How much will the rope for the 4 guys cost at 2.5^ a foot, 20' being allowed for knots? 24. What is the area of the square whose corners are at the stakes? 326 RATIONAL GRAMMAR SCHOOL ARITHMETIC 25. Find Ihe length of the diagonals of a rec- tangle 48' X 64' long. /i^ 26. A railroad runs diagonally across a rectan- FiouEE 2-:4 gulap- farm 120 rd. x 160 rd. If each rail is as long as a diagonal of the farni, how many yards are there in "the rails? 27. The center pole of a circus tent is held by 6 ropes tied to stakes at the corners of a regular hexagon (Fig. 44, p. 104). The ground is level, the pole is 48' high and a side of the hexagon is 64'. How many feet of rope are needed, if the knots take ISO'? 28. The sides of an equilateral triangle are 20' long. A straight line drawn from any vertex to the middle of the opposite side is perpendicular to this side and bisects it. What is the area of the triangle? figure l2.i 29. One side of a regular hexagon is 36'. Find the area of the hexagon? 30. A straight line drawn from the vertex of an isosceles triangle to the middle of the , base is perpendicular to the\ FIGURE 226 base. Find the area of a regular pentagon whose sides are 24", if the radius of the circular drawn around the pentagon is 18". figure 227 §189. Cubes and Cube Boots. 1. What is the volume of a cube whose edge is 5"? 6"? 7"? 11"? 18"? 21"? 24"? Definition. — The cube of a number is the product obtained by using the number 3 times as a factor. Thus the cube of4is4x4X4 = 4' = 64. Table of cubes of numbers to be learned: Cubes or Units Cubbs op Tbnc P= 1' 10' = 1000 2^= 8 20' = 8000 3^= 27 30'= 27000 43= 64 40^= 64000 5^ = 125 50=* = 125000 6^ = 216 ; 60' = 216000 7» = 343, 70' = 343000 8» = 512' 80' = 612000 9» = 729; 90' = 729000. CUBES AND CUBE R00T8 327 Definition. — The cube root of a number is one of^its 3 equal factors. Cube root is indicated by the sign y^ , Thus, i/^729 means one of the three equal factors of 729, which is 9. The sign ^/^ is called a radical sign. 2. Between what two whole numbers are the following cube roots : u^lO? ^/35? ym ^^450? ^/075? ^895? ^^ 58000? ^^ 480000? 3. Find by multiplication the values of the following cubes: (3.5)»; (6.7)»; (13.5)»; (i)'; (|)»; (|)»; (^V)'; (33J)»; (ISf)'. 4. Make a rule for finding the cube of any common fraction. 5. Prove the following relations by multiplication: ^2m = 13; ^4096 = 16; ^M^h ^^ = t\; ^IIH = H; ^27a»=3a. G. Make a rule for finding the cube root of any common fraction. 7. Find the cube roots of the following: 4; II; m; -eWV. iV«V; TiWinr. §190. Triangles Having the Same Shape (Similar Triangles). 1. Write the numerical values of the follow- . ing ratios from Fig. 228: ^^^AC- ^^^BO^ ^^^AB^ (^>^.' ^^^AO' («)^' (^)T6'' (^)t^= («)j^- ♦ The length of ab is l^^ = V^^ Why? See Prob. XII, p. 320. Figure 22« 2. In Fig. 229 the triangles have the same shape. Write the values of the ratios : ■ 1 3 i^ ^ V K* ^ u- ^ J ' f Ih ^L a f =="- s -;fe:: S^ T ?* t ^ ^- =:;2^::-: ^ _; , ^ ' BO P).-?^r (3) 3. In Fig. 230 the triangles have the same shape, and AC = 7 X ac. Find the sides of the triangle ABC^ if ac= ^V; ab = r, and *c = if'. ^ B ^'^ -^ ^^ ^^^ — ' ,^ ]^ 4r J 1^^ L "^!^ ^4~-^^c- Figure 229 Figure 230 328 KATIOKAL QKAMMAR SCHOOL ARITHMETIC 4. Supposing that ac {^^\ AC) represents 7' (Fig. 230), al, 10' and Ic^ 11', what lengths do AC^ AB, and BC represent? 5. In Figs. 231 and 232,AB = ^ab; if ^^ represents 1 mi., ^C, 3 mi., and -4C, 3J mi., what distances do ab, hc^ and ac represent? , b B j/ c ^ r:r: ■** c ^ / n V FiGuaE 231 Figure 232 6. Draw a triangle having sides of 1^', f", and i'\and another having sides of 3", 2 J", and IJ". Call the angles opposite the sides 1", f", and J", a, J, and c, respectively, and the angles opposite the sides 3", 2 J", and IJ", A^ B^ and C, respectively. Do these triangles have the same shape? Carefully cut out the triangles and place angle a over angle A ; then angle h over angle B\ and, last, angle c over (7? What do you find to be true in each case? Definition.— In triangles having the same shape, angles lying oppo- site proportional sides in different triangles are called corresponding angles. It. Read the pairs of corresponding angles in Figs. 228, 229, 230, and 232. 8. In triangles having the same shape, how do corresponding angles compare in size (see Problem 6) ? 9. In Fig. 228 find the ratio of the side lying opposite the angle B in triangle ABO to the side lying opposite the corresponding angle in triangle abc. Find a similar ratio between any other pair of such sides. How do the ratios compare? 10. Answer similar questions for the triangles of Figs. 229, 230, and 232. Definition.— In triangles having the same shape, sides, as AB and ah (Fig. 232) or BC and 6c, that lie opposite the equal angles are called corresponding sides. 11. In triangles having the same shape state a law of relation of corresponding sides. USES OF SIMILAR TRIANGLES 329 FlOURS 233 12. Triangles ABC and ade (Fig. 233) have th' same shape. How do their corresponding anglei compare? What relation holds for their cor- responding sides, i.e., how do these ratios compare : ABiad? BC'.de? ACiae? With a protractor measure each pair of corresponding angles. How do they compare in size? 13. What is the ratio of the areas of triangles ABC Skud ade (Fig. 233)? Hotv may this ratio he found from the ratio of a pair of corresponding sides, as AB and ad? 14. Draw a pair of triangles, one having sides of 1", 1^", and If", and the other having sides of 4", 6", and 7". Give the ratio of each pair of corresponding sides; the ratio of the areas of the triangles? How may the ratio of the areas be computed from the ratio of a pair of corresponding sides? 15. State a law connecting the ratio of the areas of triangles having the same shape with the ratio of any pair of corresponding Bides. §191. Uses of Similar Triangles. HOUSE -tB In Fig. 235 MNQR represents an 18"x20" board, provided with a mov- ing arm, OE^ and a protractor. The arm is made of light wood. A _^N slot ES is cut away and a thread is stretched on its under side and fastened at the ends. The ends of the thread are sunk into the wood so as to permit the thread to move smootlily over the protractor as the arm CO is turned round 0. A common pin stuck ver- tically at C and at completes the apparatus. The pin at should be driven through far enough to fasten the arm to the board at as a pivot.. The protractor is held by thumb tacks at 7, T. The board may be placed on a window sill, or on a post in a fixed position, while measure- ments are being made. It is desired to measure the distance from A to B^ A and B being on opposite sides of a large building. Use the proportion form of statement in all problems, calling the length of the unknown line x. Then find the value of x. Figure 234 Figure 235 330 RATIONAL GRAMMAR SCHOOL ARITHMETIC Solve some problems like this from your own measures. Build- ings, steeples, hills, trees, furnish good problems. 1. The board was placed on a window sill and the arm set in such position, FO^ that when the eye sighted along the pins, and />, these pins were in line with B, The mark on the pro- tractor at G was 62°. The board being held firmly in place, the arm was swung around until the pins at and at C were sighted into line with A and the reading on the protractor at H was 114°. How many degrees were there in the angle EOFt 2. The distances from A and from B to the window at were measured and found to be 484' and 520'. A triangle O'ef (shown in Fig. 234) was drawn having angle eO'/=62°, and 0'e = 4.84", O'/=6.20". The side e/ was then drawn, measured, and found to be 5.18". How long is the line ^^? 3. A woodman desires to fell only such trees as will funiish two 10-ft. cuts between the stump and the first limb. He wishes to allow 4 ft. for height of stump and waste in cutting at the bottom and top ends. To test a standing tree, he places a 4-ft. stick QR vertically in the ground 33 ft. from the tree, lies down on his back with his feet against the stake, and sights over the top of the stake to the first limb B, The distance from his eye P to the soles of his feet Q being h\ feet, should he fell the tree? 4. In Fig. 236, if P^ = 6 ft., 72^ = 41 ft., and RA = 60 ft., how long is ^i?? 5. How long is AB if PQ and RQ are the length, as in problem 4, and RA = 90 feet? 6. A woodman steps off a distance of 9 steps from a tree, faces the tree, and holds his axe-handle at arm's length in front of his eye, as the ruler is held in Fig. 125, p. 189. His arm is 28 in. long, and he finds that 2 ft. of the axe-handle just cover the dis- tance from the ground to the first limb of the tree. If his step is 3 ft. long, how high is the first limb? FIGUBE 236 COMMON FRACTIONS 331 7. If.4^ = 35rd. KD (Fig. 237)? ^ A'= 80 rd., and BV =^ 56 rd., how long is 8. Make measures on objects in your vicinity and solve such problems as are suggested by those given. 9. How long is x {Fig. 238)? 10. The thumb is held 2 in. ^1 ""^■"-:rr=x=:^:^> from the end of the pencil, and the L- ."f pencil is held 2 ft. in front of the figure 238 eye and parallel to the line to be" measured. When the end of the pencil is sighted into line with the corner i of the table, the ^ FlOUBB 239 Table end of the thumb is in line with tha corner a. How long is the end, ab, of the table, if the eye is 36 ft. from the table? 11. The eye sights past a point A over the point P on the fence (Fig. 240), and sees the upper edge of the moon in line with A and P, Then moving the eye 2^ in. up the stake, the lower edge of the moon is in line with B and P, If the stake is 20 ft. from the fence and the moon is 240,000 miles away, how long is the moon's diameter in miles? 12, Make measures like these yourself. Figure 240 332 RATIONAL GRAMMAR SCHOOL ARITHMETIC 13. The rays of light from the sun passing through a pin-hole in the screen at H ,(F^S' ^^^) give a small circular image of the FIGUBB 241 sun at /. With distances and diameter of image as shown in the cut, what is the diameter of the sun? 14. A small hole, H^ in a window screen, gave a round image of the sun, 1.1 in. in diameter on a sheet of paper held 10 ft. from the pin-hole. If the sun is 93,000,000 mi. away, what is the sun's diameter? 15. Fig. 242 shows a crude apparatus for finding distances that cannot bo measured directly. It consists of a board about 25" long and G" wide with end pieces about 6" high. The wood is cut Figure 242 iway from the end pieces so that the eye may sight through between the straight edges of two visiting cards at B^ past two parallel threads at T, set carefully ^" apart and 25" in front of £. • If S is in line with the upper thread of the instrument and D with the lower thread, how far is it from the stake at SD to the instrument, if SD = 8'6"? SG Suggestion. — J : 25 = 8i : x. Then, ' Ja? = 8i X 25, or -g = 212J. Find X. 16. A pupil sighted through J5' at a stake held by another pupil at SD' When the pupil at E sighted over the top thread, the USES OF SIMILAR TRIANGLES 333 pupil had to put his hand at S to line in with the slit at B and upper thread at T. The pupil at E then beckoned him to slide his hand down the pole until it lined in with the lower thread at T, The distance SB was 7.52'. IIow far was it from the instrument to the flagpole SB? 17. The distance from F to the bottom of the building, Fig. 243 Figure 243 is 288', the distance FB = 8' and AB = 18"; how high is the building? 18. In Fig. 244 it is desired to find the distance across the lake from C to B. A surveyor stuck a flagpole at a place from which he could see both and B. He measured the distances from B and from C to the pole he is holding, and found them to be 4563' Figure 244 and 5481' respectively. He then set a pole at B, j\ of the dis- tance from the first pole to By and another pole at Ay -j^q of the distance from the first pole to C. The distance from A to B was measured and found to be 84.64'. How far is it from to B? 334 RATIONAL GRAMMAR SCHOOL ARITHMETIC 19. It is desired to make a scale drawing of the tract of ground ABCDEF (Fig. 245). The parts of the apparatus to be used, as shown at the right, are a square board about 16"xl6", with a B m FiGUBE 245 block containing a 1" hole to fit over the top pin of a sharp stake to be stuck in the ground to support the board. A small level (which may bo a phial of water) and a foot rule with a vertical pin at each end for sights to be used for sighting complete the apparatus. The board is set up in the field as shown, a sheet of paper is pinned on it with thumb tacks, and the foot rule is placed upon the paper. A third pin is stuck near the center of the board at a point (not shown in Fig. 245). Holding the edge of the foot rule against this center pin, sighting along the pins toward a pole at A the observer turns the front of the foot rule until the two pins of the ruler are in line with A. He holds the ruler and draws a line along its edge on the paper. He now holds the edge of the ruler against the center pin, and carefully sights the two ruler pins into line with B^ and, hold- ing the ruler in place, draws a line on the paper toward B. He proceeds in the same way with each of the points C, D^ E^ and F^ being careful not to turn the board around on the stake pin. 20. The lines from the stake supporting the board to ^, to J5, to C, and so on to ^were measured and found as follows: to A^ 460'; to B, 452'; to C, 378'; to Z>, 527'; to E, 535' and to F, 832'. Using a scale of 1": 100', the distance to A (460') was laid USES OF SIMILAR TRIANGLES 335 off from the center pin on the line drawn toward A giving a (on the board), the distance to B (462') was laid off from the center pin on the line drawn toward B giving b (on the board), and so on around to F. Calling the center pin o, how long is oa? ob^ oc? od? oe? of? 21. With a ruler the points a, S, c, d, e and / were connected as shown in the figure. The lines were ab = 2.8"; be = 4.5"; cd = 5.12"; de = 2.95"; 6/= 6.25"; fa = 4.48". How long are the lines AB, BC, CD, DE, EF, and FA? The board may be supported by a light camera tripod or by a home-made tripod and the board may be held to the flat top of the tripod by a thumb nut. (See Fig 246.) 22. The distances from the apparatus to the corners of the field (Fig. 246) were: to J, 678'; to B, 612'; to (7, 683'; to D, 738'; ..^^mm--^^ Figure 246 to E, 698'; to F, 625'; to G, 679'. Using a scale of 1":200', how long should the distances be made from the center pin to a? to b? to c? to d? to e? to/? to g? 23. With a ruler a, 5, c, c?, e, /, g, and a were then connected and lines measured. The measures were: ab= 3.81"; Sc = 4.12"; ceZ = 2.86"; e?c = 2.75"; e/=5.86"; /(7 = 6.18" and ^fl = 5.98". How long are the lines AB? BC? CD? DE? EF? FG? GA? 33G RATIONAL GRAMMAR SCHOOL ARITHMETIC 24. Having made the scale drawings, perpendiculars may be drawn from the center pin to each of the pides ab^ J^ and so on. From the measured lengths of these perpendiculars and the bases AB, BC^ and so on of the triangular parts of the figures, the areas of the triangles aoh^ boc, and so on (calling o the point where the center pin stands) may be computed. The sum of these areas gives the total area of the figure abcdefga, 25. Suppose the area of triangle aob were 4.94 sq. in. what would be the area of A OB ( being the point on the ground just under o), if the scale were 1":200'? (See Problems 13 and 15, §190). 26. If the areas of the triangles aoby boc^ cod^ doe^ eof^ and foa (Fig. 245 prob. 7) in square inches were: 6.073, 7.740, 9.172, 7.490, 16.725 and 7.563, respectively, and the scale were 1": 100'; what were the areas of the triangles AOB^ BOC^ COD^ DOE^ EOF, and FOA'^ AFFLIGATIONS OF FERGENTAOE §192. Insurance. 1. What will it cost to insure $680 worth of household furni- ture, at the rate of If %? Definition. — The amount paid for insurance is called the premium. 2. An art gallery, valued at $500,000, is insured at the rate of li%. What is the premium? Definitions. — The written agreement between an insurance company and the insured is called a policy. The amount for which the property is insured is the /ace of the policy. 3. A vessel, valued at $16,000, was insured for f of its value. Find the face of the policy. 4. A growing crop was insured at 5%. ' The premium was $140. What was the face of the policy? 5. If it costs $420 to insure a house for J of its value, at 3.J^%, what is the house worth? APPLICATIONS OF PERCEXTAGE 337 6. A stock of hardware was insured for 17000, insurance cost- ing 1110.75. What was the rate ot insurance? 7. If a shipment of grain is worth 1840, and the premium amounts to $17.60, what is the rate of insurance? 8. A machinist insured his tools, valued at $360, for i of their value, .paying $8.19. What rate did he pay? 9. A farmer paid a premium of $10.50 for insuring his stock at 1^%. For what amount was the stock insured? 10. A man paid $67.50 for the insurance of a steam launch at 1^%. For what amount was the launch insured? 11. What is the rate paid for insuring a bridge, valued at $15,000, for f of its value, the premium being $240? 12. A library, worth $28,000, was insured for f of its value, the premium being $720. What was the rate of insurance? 13. The contents of a grain elevator were insured at a rate of f %. What was the amount of the policy, the premium paid being $272.40? 14. The premium paid for insuring a quantity of lumber, for two-thirds of its entire value, at 3 % , was $36. What was the value of the lumber? 15. A piece of property, valued at $27,200, was insured fori of its value. The premium was $212.50. What was the rate? 16. A tank of oil, holding 2592 gal., worth 18^ per gallon, was insured at 4%. Find the premium. 17. A man insured a cargo worth $2400 at 3^%. f of the cargo was lost at sea. Wliat amount of insurance should the man obtain? What premium had he paid? 18. What is the rate of insurance paid for insuring 26,000 bu. of grain Vorth 77^, for f of its value, if the premium is $500.50? 19. What sums must be paid for policies to insure a factory, valued at $21,000, at If % ; a house, worth $28,000, at i% ; and a barn, valued at $5600, at If %? 20. A steamboat had a cargo, valued at $2000, which was destroyed by fire. The insurance was $1500. What per cent of the value of the cargo was covered by insurance? What was the premium, at 4^%? 338 RATIONAL GRAMMAR SCHOOL ARITHMETIC §193. Taxes. Definition. — A tax is a sum of money levied by the proper officers to defray the expenses of national, state, county, and city governments, and for public schools and public improvements. 1. A town wishes t.o raise 123,500 to build a public hall. If the collector's commission is 2|%, what is the total amount to be collected? 2. If the taxable property of a town is valued at $923,846 and the rate of taxation is 3f %, what is the whole amount of the tax? ' 3. The taxable property of a town is $869,472, and the rate is 8^ mills on the dollar. What is the amount of the tax? 4. What amount of money must be collected to raise a net tax of $643,000, allowing 6% for collection? 5. The tax in a city is $49,682; the rate is 15 mills on the dollar. What is the assessed valuation? Definition — Assessed valuation means the estimated value of the property that is assessed. 6. What IS the rate of taxation when property assessed at $8940 pays a tax of $160.92? 7. The net amount collected as a. tax was $2896.42. The collector's commission was 3^%. What was the whole amount collected? 8. The assessed valuation of the taxable property of a town was $1,218,694. The tax to be raised was $18,280.41. How many mills on the dollar was the rate of taxation? 9. A tax collector received $175.18 as his 2% commission on a certain sum collected. The assessed valuation of the taxable property was $922,000. What was the tax of a man, whose prop- erty was valued at 118,000? 10. The real property (houses, lands, etc.) of a certain town is valued at $4,560,800, and the personal property (movable property) at $945,900. If $12,000 is to be raised by taxation, how much must a man pay, whose property is valued at $9,840? 11. A net tax of $8965 is to be raised in a certain city. The APPLICATIOXS OF PERCEKTAOE 339 assessed valuation of the taxable property is $5,689,243. The collector's commission is 1^%. What will be the tax of a man, whose property is valued at 156,000? 12. Property on a city street is assessed 2% on its valuation. How much more will a man pay who owns 100 front feet, valued at $125 per foot, than one who owns property valued at $7500? 13. If the assessed valuation of the taxable property in a cer- tain city is $1,069,210, and the whole amount of tax collected is $13,899.73, what is the rate of taxation and the net amount after deducting the collector's commission of 1|%? 14. The taxable property in a town is $872,990. The rate of taxation is .015. What is the net amount collected after deduct- ing the collector's commission of 2^%? 15. The assessed valuation of the taxable property in a certain city is $4,968,390. The tax collected amounts to $94,399.41. How many mills on the dollar was the rate of taxation? 16. A certain town wishes to raise by taxes $20,500 to build a schoolhouse. What tax must be levied to cover this and the cost of collection at 4%? §194. Trade Discount. 1. A hardware dealer bought the goods named in the follow- ing bill : (1) 2 doz. braces, @ 45^ each; (2) 50 1b. J" bolts, @ 3i^; (3) 12 bales barb wire, @ $2.15; (4) 25 kegs nails, @ $2.50; (5) 8 cooking stoves, @ $28.75; (6) 12 baseburners, @ $32.50; (7) 150 lb. screws, @ $4.75 per cwt. The bill was subject to discounts of 10%, 7%, and also 5% 30 da. or 6% 10 da. What was the total cost if the bill was paid in 30 da.? in 10 days?, 2. Find the amount needed to settle the following bill, sub- 340 . RATIONAL GRAMMAR SCHOOL ABITHMETIC ject to the successive discounts, 10%, 8%, and 5% off for 30 da. or C% for cash, the bill being paid in cash: (1) 2 chests carpenter's tools, @ $48.50; (2) 2 doz. augurs, @ 25^ each; (3) 2 doz. carpenter's rules, @ $1.80; (4) 480 lb. hinges, @ $4.75 per C; (5) 3 doz. tin buckets, @ 22^ each; (6) 4 doz. gardening rakes, @ 28^ each; (7) 14 scoop shovels, @ 85^; (8) 6 wash boilers, @ 78^; (9) 12 washtubs, @ 52^; (10) 10 doz. clotheslines, @ $1.10. 3. What amount will be needed to settle the bill of problem 2 in 30 days? 4. What amount will settle the following bill of house furnish- ings, the bill being subject to the discounts, 20%, 10%, 8%, and 5% 30 da., 6% cash, the bill being paid in 30 da.? (1) 8 dining tables, @ $30; (2) 12 sets dining-room chairs, @ $18.50 a set; (3) 10 bookcases, @ $12.50; (4) 8 rocking chairs, @ $12.75; (5) 15 center tables, @ $15; (6) 12 Wilton rugs, @ $25; (7) 15 Axminster rugs, @ $23; (8) 250 yd. ingrain carpet, @ 45^. 5. What amount paid in cash will be needed to settle the bill? 6. What amount in cash will settle this bill of plumber's supplies : (1) 40' lead pipe, @ 15^, discounts 20%, 10%, 7%, 2 f^ cash; (2) 100' iron tubing, @ 3^^, discounts 20%, 10%, 7%, 2% cash; (3) 8 bathtubs, @ $20, discount's 20%, 7%, 2% cash; (4) 8 lavatory outfits, @ $10, discounts 20%, 7%, 2% cash, (5) water meters, @ $5, discounts 20%, 12%, 2% cash; {()) 4 pumps, @ $25, discounts 20%, 7%, 2% cash; (7) 5 pump valves, @ $2.85, discounts 20%, 7%, 2% cash? APPLICATIONS OF PERCENTAGE 341 §195. Stocks and Bonds. A Stock is a written agreement (called also a stock cerlificate)^ made by a company to pay the holder a certain part of the eara- ings of the company. The sum paid is reckoned at a certain number of dollars per share^ or at a certain rate per cent of the face value of a share. A share usually has a face value of $100, $500, $50, $1000, etc. A share of mining stock often has a much smaller face value. When a stock company pays to the holders of its stock $2 on every $100 of its capital stock, or 2% on its stock, the company is said to be paying a S^ dividejid^ or a ^% dividend. When stock is paying a high rate of dividend it may sell in the market above par^ or for more than its face value. If the rate of dividend is low, the stock may sell for less than its face value, or below par. When it sells for its face value it is sold at par. city government, or by a company, to pay the holder a stated sum of money with interest on it at a stated rate. The sum of money to be repaid is called the face of the bond. Stocks and Bonds are usually purchased tHrough an agent, called a broker^ who makes a business of buying and selling stocks and bonds. He charges a certain per cent (usually i%) of the face value for buying, and an equal rate for selling. This charge is called brokerage. In the following problems, regard the face value of the stock or bond as $100 and the brokerage as ^%, unless otherwise stated. Following is a list of newspaper quotations on the N. Y. stock market on certain stocks and bonds : STOCKS Open Amal. Copper. 9lb% American Sugar 129^ B. &0.com 101^ B. &0. pfd 9554 Chi. & Alton com. . . . 36J4 Chi. & Alton pfd 71 Vi 111. Central R. R 148 Peoples Gas \Qlb% Pressed Steel com . . . 65Vi Pi-essed Steel pfd. . . . 94Vi Rubber Goods com.. 24 V4 U.S. Steel com ^\ U. S. Steel T)fd 87% Si. L.. and S. F. com. 78H St. L. and S. F. pfd. . 79% No. Sales 27000 Atchison Gen. 4s. 12000 B. and O. gold 4s.. , 1000 B. andCSHs 1000 C. B. andQ. 4s POOO'C. and A. S^s 28000 U.P Is HiRh Low Noon 66Ji 9ib% es>% 130 129^ 129S1£ 101^ 1005^ 101 95^ ^5Vi 95H 37 86^ 36^ 72^ 71>i 72 148^ 148 148^ 105% 105^ 105% eSH 66J^ 651/3 941/a 9414 94H 2A% 241/, 241/s m\ 37H 37H 87H 81^^ 87M 90% 78 80J^ 80/8 rdJi 80J/a BONDS Closing bid and asked prices for govern- ment bonds were as follows: nid Asked New 2s 1 08 -^^ 1 09 14 Coupons It 8^ 109J^i New 3s IO8M 109 New 3s coupon 108?^ 109 New 8s small 108 109 Registered 4s 112^ 113 Coupon 4s . . 112H 113 Registered 4s new 139J^ 1395^ Coupon 4s new 1393i 139% Registered 5s 107^ \m% Coupon 58 107J4 10796 BOND TRANSACTIONS .100H@100^ .101 @101H . 94H . 106^^ 76H@ 76?^ .102>^ No. Sales 5000 U. S. Leather 4s 85J^ 3000 B. «& O. coupon 4s 106M@1063g 82000 Cen. Pac. R. R. 8Hs 88k <S) 8«?% 10000 Wabash R. R 5.s 11r>k(^115i5 ICOOO Manhattan 4s 105^ 343 RATIONAL GRAMMAR SCHOOL ARITHMETIC 1. A certain express company has a capital stock of $500,000, divided among its stockholders in shares of the par value of $100 each. In 6 mo. the net profits amount to $10,500, which the company distributes among its stockholders as a dividend. What is the rate of dividend? How much does a holder of 200 shares receive as dividend ? 2. What must a man have paid for 400 shares AmaL Copper stock on the date of the table, including brokerage, if he bought at the opening price? at the highest price? at the lowest? at the noon price? 3. Answer similar questions for U. S. Steel pfd. (preferred). Definitions. — Preferred stocks are stocks which pay a fixed dividend (say of 7%) before any dividends are paid on common stocks; Common stocks pay dividends dependent on the net earnings of the company after expenses and dividends on preferred stock have been paid. 4. How much did a man gain or lose by buying 2000 Am. Sugar at the opening price and . selling at the highest price, paying brokerage of i% for buying and also for selling? 5. Answer similar questions for B. & 0. com. ; for B. & 0. pfd. 6. How much does a man make or lose who buys 2500 shares 111. Cen. R. E. stock at the lowest quotation of the table and sells at the highest, paying brokerage of i for buying and ^ for selling? 7. If a man invests in St. L. and S. F. pfd. stock, paying 7% annual dividend, at the lowest quotation of the table, what interest does he receive on his investment? Suggestion.— The investor must pay |79| + IJ per" share (of $100), and he receives ^7 per year as interest. 8. Answer a similar question on C. & A. pfd. 9. What interest does a man receive on an investment in Gov- ernment Xew 2s if he invests at the price "bid," brokerage ^? at the price "asked"? Note. — Government 2s, 3s etc., are government bonds paying 2% 3fo, etc., annually. 10. Answer similar questions for New Ss; for New 3s cou- pon ; for Eegistered 4s ; for Registered 4s new. 11. What rate of interest does the investor who bought the 1000 B, & 0. 3^8 receive? APPLICATIONS OF PERCENTAGE 343 12. Answer similar questions for the investor who bought the 9000 C. & A. 3^8, if he paid the lowest quoted price; the highest. ] 3. Which pays the higher rate of interest on the investment, and by how much, U. P. 4s as quoted, or Wabash R. R. 5s at ihe lowest quotation? at the highest? 14. Which is the more profitable investment, and by how much, B. & 0. coupon 4'8 at the lowest quotation, or a straight loan at 4%? 15. Answer other similar questions on the table. 16. An investor in Chi. & Alton com. stock at the highest quotation received two 2% and one 3% dividend, and sold the stock 18 mo. later at 38^. What rate of interest did ho receive on his investment? How much profit did he make if he bought. 1000 shares? 17. Answer similar questions on the table. §196. Compound Interest. With certain classes of notes, if the interest is not paid when due, it is added to the principal and this amount becomes a new principal, which draws interest. Savings banks add the interest on savings deposits at each interest-paying period and pay interest on the entire amount. This interest on interest is called com- pound interest. Bankers also collect their interest on loans and then reloan the interest, and in this way virtually receive com- pound interest on their money. Compound interest is usually payable annually, or semi- annually. 1. Find the compound interest on a note of $200 at 6% for 5 yr., interest payable annually. Solution.— Interest on 1200 for 1 yr. at 6% = 200 X $ .06 = % 12.00 Amount of $200 for 1 yr. at 6% = 200 X S 1.06 = $212.00 Amount of $212 for 1 yr. at 6% = 200 X $(1.06)2 = §224.72 Amount of $224.72 for 1 yr. at 6% = 200 X $(106)3 = $238.23 Amount of $238.23 for 1 yr. at 6% = 200 x $(1.06)* = $252.52 Amount of $252.52 for 1 yr. at 6% = 200 X $(1.06)^ = $267.68 The compound interest=$267. 68— $200=$67. 68. Note. — The $267.68 is called the compound amount The expressions (1.06)', (1.06)*. and (1.06)6 mean 1.06 X 1.06 X 1.06, 1.06 X 106 X 106 X 1.06, and 1.06 X 1.06 X 1.06 X 106 X 106. and are read ''1.06 cube," **1.06 to the 4th power," and "1.06 to the 5th power." 344 RATIONAL GRAMMAR SCHOOL ARITHMETIC 2. Find the simple interest on a note for the same face at the same rate and for the same time as in problem 1. By how mnch does the compound interest exceed the simple interest? 3. Find the compound interest on $180 at 6% for 3 yr., interest compounded semi-annually. Solution.— Amount on |180 for J yr. at 6% = 180 X 1 1.03 = $185.40 Amount on $185.40 for J yr. at 6% = 180 X 1(1.03)2 =^ $i90.96 Amount on 1190.96 for J yr. at 6% = 180 X $(1.03)' = $196.69 Amount on $196.69 for J yr. at 6% = 180 X $(1.03)* = $202.59 Amount on $202.59 for i yr. at 6% = 180 X $(1.03)^ = $208.67 Amount on $208.67 for J yr. at 6% = 180 X $(1.03)» = $214.93 Compound interest = $214.93 — $180 = $34.93. 4. If P denotes any principal at r% for n yr., interest being compounded annually, show that if A denotes the compound amount, -( ry 1 + 100/ 5. Show that if the interest is compounded semi-annually, the letters meaning the same as in Problem 4. 6. Show^ that if / denotes the interest, compounded annually, 7. If the interest is compounded semi-annually, show that 8. A man has a deposit of $100 in a savings bank paying 4%, interest compounded semi-annually, for 3 yr. How much is due tlie depositor at the end of that time? 0. How much is due on a savings deposit of $250, in a savings V)ank paying 3% semi-annually, the deposit having been in the bank 3^ years? USB OF LETTERS TO REPRESENT NUMBERS 345 10. A man paid compound interest at 6% , interest compounded annually, on a note of $150 for 7 yr. How much was required to pay the note? Note. — Many notes have coupon notes attached for the amount of interest due at each interest-paying period. These coupon notes usually bear a higher rate of interest than does the note itself. 11. A note for $500, bearing 6% interest, has 3 interest coupons attached, the coupons, when due, bearing 7% interest, payable annually. If the note runs 4 yr., no coupons being paid in the meantime, how much money will be required to pay the entire debt at the end of this time? 12. A note of $450, bearing interest at 7%, interest pay- able semi-annually, coupons bearing 8% interest, amounts to what sum at the end of 4^ yr., no coupons being paid until the end of the time? USE OF LETTERS TO REPRESENT NUMBERS §197. Problems. 1. Joseph had 45^ and he earned 15^ more selling oranges. IIow many cents had he then? (Answer by indicating the opera- tion you perform, thus: 45^ -f- 15^.) 2. William had x marbles and Harold gave him y marbles. How many marbles had he then? 3. A cow gave x lb. of milk at the morning milking and y lb. at the evening milking. How many pounds did she give at both? 4. James earned 80 jJ selling papers on Saturday and spent 45^ of his earnings. How many cents did he save on Saturday? (Answer by indicating the operation you use.) 5. During July a boy earned m dollars and spent s dollars. How many dollars did he save? 6. Helen bought 15 pencils at 3^ apiece. How much did she pay for all? 7. Elizabeth took x music lessons during February at y dollars per lesson. What was the cost of her lessons for the month? 346 RATIONAL GRAMMAR SCHOOL ARITHMETIC 8. A thermometer rose ^° on one day and 5 times as much the day following. How much did it rise on the following day? 9. The area of a rectangular lot is 63 sq. rd. and the length of one side is 9 rd. How long is the other side? 10. The area of a rectangle is x square inches and the length of one side is \j inches. How long is the other side? 11. How many lots each of h ft. frontage can be made from a frontage of a feet? 12. An orange boy earns a cents on Wednesday, three times as many cents on Thursday, and as many cents on Friday as on Wednesday and Thursday together. On all three days he earns 80^. How. much does he earn on Thursday? on Friday? Note. — In all such problems use the equation. We have a + 3a -|- 4a = 80, or 8rt = 80. If 8a = 80, a = 10. 3a = 30, Thursday earnings, and 4a = 40, Friday earnings. 13. The altitude of a rectangle is a; in. and the base is Zx in. How long is the perimeter? What is the area of the rectangle? 14. 4a means 4 x a. Compare the values of a x 4 x a and 4 x a X a. How is a X a written? How is 4 x a x a written? Ans, 4a". 15. The mercury stood at x degrees at 2 p.m. and fell 3° dur- ing the next hour. The reading at 3 p.m. was 25°. What was the reading at 2 p.m.? 16. The mercury stood at 12° at 8 a.m. During the next two hours it rose x degrees. What was the thermometer reading at 10 a.m.? If it had fallen y degrees, what would have been the reading at 10 a.m.? 17. The thermometer read 28°, the mercury fell x degrees, then Zx degrees, and then rose ^x degrees, when the reading was 32°. What was the number of degrees in each o? the three changes? 18. James paid %x for a hat and twice as much for a coat. He paid $4.50 for both. What did he pay for each? 19. I paid 130 for a bicycle and sold it for %x. How much did I gain? 20. I paid ^ of what I gained for a coat. How much did I pay for the coat? USB OP LETTERS TO REPRESENT NUMBERS 347 21. Louis had x apples and aj^e y of them. How many did he have left? 22. Henry had 7a papers and sold 4a of them. How many did he have left? 23. I walked x miles due south one day and y miles duo north the next day. How far was I then from my starting point? 24. A man rows a boat downstream at a rate that would carry his boat a miles per hour through still water, and the current alone would carry him down h miles per hour. How far will he go in 1 hour? 25. A man walks from rear to front through a railway coach 3 mi. per hour, and the coach is running at the same time a mi. per hour. How fast does the man pass the telegraph poles along the track? 26. How fast does he pass them if he walks from front to rear? 27. Mary had a pencils and sold them at 5^ apiece. How many cents did she receive for them? 28. James sold x oranges at a cents apiece. How many cents did he receive for them? 29. A dealer sold a wagons for 40a dollars. What was the price per wagon? 30. A farmer paid x dollars for 15 A. of land. How much did he pay per acre? 31. The area of a rectangle is m sq. rd. and it is I rd. long. How wide is the rectangle? 32. A township is x mi. square and contains 36 sq. mi. What is the value of a:? What does x represent? 33. Helen bought x dolls at 10^ apiece and y yd. of muslin at 8^ a yard. How many cents did she pay for both? 34. James had m cents and earned c cents more. He invested all his money in papers at 3^ apiece. How many papers did he buy? 35. He sold his papers at 5^ apiece. How much did he receive for the papers? How much did he gain? 36. The base of a triangle is x ft. and the altitude is y ft. What is the area in square inches? 348 RATIONAL GRAMMAR SCHOOL ARITHMETIC 37. The area of a triangle is a; sq. ft. and the base is 6 ft. long. What is the altitude? 38. A parallelogram has a base {x + y) in. long, and is 6 in. high. What is the area? Express this answer in two ways and make an equation by writing the two expressions equal. Why are the two expressions equal? §198. Uses of the Equation. 1. If each brick weighs 7 lb. and there are 10 bricks in the bucket (Fig. 247), with how many pounds of force does the bucket pull downward on the rope, the bucket itself weighing 5 pounds? 2. If we denote the number of pounds of force with which the man pulls downward on the rope to balance the bucket by p, write an equation showing the number of pounds in p, the 5-lb. bucket being loaded with 10 brickg. 3. A horse is raising a ^ 10-ft. steel I-beam weighing FiGURB 247 30 lb. for each foot of length. If the force the horse must exert to hold the beam suspended in the air be denoted by JP, write an equation showing the number of pounds in F. 4. A wagon weighing 1800 lb. is loaded with 3 T. of coal. When it is being drawn ^^''''^^ ^^® over ordinary pavement it pulls backward on tbe traces of the team with a force of ^V ^^ many pounds as there are pounds in the entire weight of the coal and wagon. If the force exerted by the moving team is F lb. , write an equation showing the number of pounds in F. 5. Write an equation showing how many pounds each horse draws if both draw with equal force. These probbms show how the equation may be used in solving simple problems of mechanics; and we need to learn the laws upon which the use of the equation is based. USE OP LETTERS TO REPRE8EKT NUMBERS 349 §199. Principles for Using the Equation. Beview p. 97, 1. Two weights, one of x lb. and the other of y lb., are tied to strings which pass over pulleys at A and B (Fig. 249). The strings are knotted, at C. If x is greater than y, how will C move? Under what condition will G remain at rest? What relation is shown to exist between x and ^ by a move- ment of C toward the right? (T's standing stationary indicates a balance %n value between the forces y lb., drawing toward the right, and x lb., drawing toward the left. This fact is expressed in symbols by writing a: lb. = ^ lb., or, better, by a: = y, simply. 2. If a 10-lb. weight be hung to the weight y, how many 5-lb. weights must be hung to a; to secure a balance of value 9 Note. — ^This is shown by writing y+10s=a?+5 + 5, read **y plus 10 equals X plus 5 plus 5." 3. How many pounds in all were added to xl 4. If 50 lb. were added on the right, how many 5-lb. weights must be added on the left before the sign of equality (=) can be written between the numbers? If 7a lb. were added on the right, how many a-lb. weights would be needed on the left for balance? 5. How many pounds would be needed on the right if b lb. are added on the left? S + c lb.? jt? + 35 pounds? 6. Write the equation form of statement for each case of Prob- lems 4 and 5. 7. If a greater number of pounds were added to y lb. than to X lb., say 15 lb. to y lb. and 10 lb. to x lb., what would occur in the apparatus shown in Fig. 249? This destroying the balance, or emmlity. Is stated briefly in signs by writing y + 15 > X + 10, which is read *'y plus 15 ia greater thanx plus 10." If the balance, or equality, is destroyed by adding a heavier weight to X than to y, the fact is stated in symbols thus: j^ + 10 < cc + 15; read **y plus 10 is less tJian x plus 15." Notice in each case that the vertex at the horizontal V always points toward the smaller number. Definitions. — An expression in which the sign < or > stands between two numbers is called an expression of inequality. 350 RATIONAL GRAMMAR SCHOOL ARITHMETIC We now have signs for writing briefly the three possible relations which may exist between any two numbers, as a and b. They are called relation signs. 8. Read and give the meanings of (1) a > J, (2) a = S, (3) « < S, (4) ic + 20 < a; + 15 -f 10. 9. It y = x^ and a lb. are added to y lb. and b lb. to x lb., how will the knot C move for case (1) Problem 8? for case (2)? for case (3) ? State in symbols the fact shown by the knot C in each case after adding the weights, using the correct relation signs. 10. If any given weight, a lb., be added to ^, what must be true of the weight b lb. if, when added to a;, we may write y-^a = x + b? 11. If z lb. be added to both the y lb. and the x lb., when y = a:, what equation may we write? Principle I (For Addition of Equations). — 1/ the same number y or equal numbers^ be added to both sides of an equatioii^ the sums' are equal. Illustration. — If y=^x, and a = 6, then we may write and y -j- a := x - - a, b = x-j-b. and y + a = x + 6, and y + 6 = a? + «. 12. If 8 lb. be removed from the left scale pan (Fig. 249), how many 4-lb. weights must be removed from the right pan to restore the balance ? Note. — This is written in signs y — 8 = a? — 4 — 4, and is read *'y — 8 equals x minus 4 minus 4." 13. How many pounds in all were removed from the right scale pan? 14. It y = Xj and if a lb. be removed from the left, and b from the right, to what must the value of b be equal to enable us to write y — a = x — b? Principle II (For the Subtraction of Equations). — If the same number^ or equal numbers, are subtracted from equal num- bers, the differences are equal. Illustration. — If y = a? and a = 6, then we may write: y — a = x — a, and y — b = x — b, and y — a = x — 6, and y — b = x — a. USE OF LETTERS TO REPRESENT NUMBERS 351 15. If in Fig. 249 y be doubled, what corresponding change in X will restore the balance? Note. — The double of y is written 2y and read "two y." 16. What equation states that there is a balantje? 17. li y = x^ and if 4 weights, each equal to y, are put on the right of the apparatus in Fig. 249, how many weights, each equal to x^ must be put on the left to keep the balance? 18. If a weights, each equal to y, are on the right, how many weights, each equal to x^ must go on the left to secure balance? Note.— The equation is ay = ax, 19. If a = S, and a weights, each equal to ?/, are put on the right, and b weights, each equal to re, are put on the left, what will be shown by the scales if y = x. (Answer with an equation,) Principle III (For the Multiplication of Equations). — If equal riumbers are multiplied by the same number^ or by equal numbers^ the products are equal. Illustration. —If y=^x and a = 6, then ay = ajx, and by = bx, and ay = bx, and ax = hy. 20. If y = Xy and half of the weight on the left be taken off, what fractional part of the weight on the right must be taken off to restore the balance? 21. If only ^ of y be kept on the right (Fig. 249), what frac- tional part of X must remain on the left? 22. If only ^ lb. be kept on the right (Fig. 249), what frac- tional part of X lb. must remain on the left for balance? 23. Suppose y = X and a = b, and that — lb. are on the right, a X and -T lb. are on the left ; what equation would be shown to be true by the apparatus? Note. — The equation is — =-r, read "y divided by a equals x divided a by 0." 352 RATIONAL GRAMMAR SCHOOL ARITHMETIC Principle IV (For the Division of Equations). — If equal numbers are divided by the same 7iunibery or by equal numbers^ the quotie?its are equal. Illustration.— If y = x and a = 6, then y_x a a* and y_x 6" 6' and y_x and y_x b a' In Fig. 183, p. 293, a {x-\-y) represents the area of the whole rectangle, while ax and ay represent the areas of its two parts. As the two parts of the large rectangle, taken together, must equal the whole rectangle, we may write : a{x + y) = ax-^ay In Fig. 184, p. 294, the area of the whole rectangle is given by (a + b) {x -f y) , while the areas of the several parts are ax, ay, bx^ and by. Since the parts, taken together, make up the whole rectangle, we may write: (a + J) (a; + y) ^ ax + ay + bx-^- by These two equations illustrate the meaning of a fifth principle of the equation, viz. : Principle V. — Any whole equals the sum of all its parts. These five fundamental principles or laws, exemplified by the scales, p. 97. and the pulley device (Fig. 249), p. 349, 7nust 7iot be violated in using the equation. §200. Problems. 1. Find the value of the letter rr, y, or z, in each of the first nine problems: (1) 3a: = 15 lb. ; (4) dz = 36i^; (7) llic = 55; (2) Sx = 24 ft. ; (5) ix = $3 ; (8) iy = 21 ; (3) 7y = 18 mi. ; (6) iy = 14 sq. in. ; (9) ax = 3a. 2. If 3a; = 9, to what is 2x equal? 7a;? |:r? 3. If 7.5 = 21, to what is 5.c equal? far? eja;? USE OP LETTERS TO REPRESENT NUMBERS 353 4. If ic + 3 = 8, what is a;? 3a;? 5a;? ic*? __ 5. If 2a; + 7 = 15, what is a;? 3a;? 7a;? ^? s/x'i 6. If a; - 3 = 6, what is xt 3a;? Q?t y/x? 7. 2x 4- 3a; + 6a; = 33; find x. 8. 4a; - 2a; + a; = 64; find a;. 9. 7a;-a; + 2a; = 32; find a;. 10. xy = 16, and ^ = 8 ; what is a;? If a; = 8, what is y? 11. ax = 35, and « = 5 ; what is x? If a- = 5, what is a? 12. ab = 10 1 what is ^ if a = 1? 2? 3? 4? 10? 20?- 13. By what principle may we write c(x + y-{'z) - cx + cy + cZj Xj y, and z denoting the bases of 3 rectangles whose altitudes are each equal to f? (Answer by sketching the proper figure and pointing out the rectangles whose areas represent each of the products in the equation.) 14. If a = 8 and & = 5, find the values of the following expres- sions and tell what ones are equal : (1) (a + h); (4) {a + by ; (7) a' - 2ah + h' ; (10) a' + ¥ ; (2) {a -b); (5) (a - bf ; (8) a' + 2ab + ¥ ; (11) a{a' + V) ; (3) ^aS; \&)a'-\-b^i {^) {a + b){a-b); {12) b{a-b). 15. If a; = 9 and y = 4, tell which of the following express true relations and write the correct relation sign in each case: (See p. 350.) (l)a:>y; (11) {x + yf^i^ + yy-, (2)a; = y; (12) {x-yf = x'-f^ (3) x<y', (13) {x-^yY = x^ + 2xy-\-y^i {'l)y<x\ ' {U) lx-yy = x^-2xy + y'^; (5) a; + y = 13; (15) {x-yf = x^ + 2xy + y^; (6) a;-y = 5; (16) {x + y){x-y)<x'-f', (7) (a; + ^)2 = 169; (17) {x+y){x-y)=x' -^2xyi^y''; (8) {x-yr = 25; (18) {x-y){x-^y) = (a^ + f); (9) a:2-jy«>25; (19) {x-y){x4-y) <{x^ -y^); (10) ar^-fy2 = 169; (20) x{x + y)^ x'-i-xy. 354 RATIONAL GRAMMAR SCHOOL ARITHMETIC §201. Statements in WorcU and in Symbols. 1. Write the symbolic statements for these verbal phrases and statements. Let x stand for the number when there is but one number to be symbolized in the problem. (1) A certain number increased by 15. (2) Twice a number diminished by 8. (3) Seven times a number increased by three times the number. (4) The square of a number, divided by 8. (5) The sum of the square and the first power of a number. (G) Eight times a number, divided by three. (7) One-third of 10 times a number. (8) Three times a certain number, diminished by one, equals 20. (9) Eighteen times the square of a number equals 72. (10) Twenty-five times a number, increased by 5, equals 30 times the number, diminished by 15. (11) One-eighth of the sum of a certain number and 18. (12) Six times the difference between a certain number and 3 equals 18. (13) The product of the sum and the difference of x and 3 equals 18 (x being greater than 3). (14) The difference between ic and 18 is greater than 18; is less than 25; is equal to 20 (x > 18 in each case). 2. State in words what these expressions mean. For example, (1) means "double a certain number, diminished by 9": (1) 2^-9; (6) (a;+l)(ar + l); (11) 2a; + 6 = 12; (2) lG.r + .»-; (7) x{x-^y. (12) 7a:-2 + 16; (3) 28a; + 17; (8) 3(9-0:); (13) 5a:+7 = 42; (4) .^ + .n (9) 12(^-1); (14)(a;-4)(a; + 4)=20; (5) .«-.<•; (10) (^-l)(x-l); (15) {a^-V){(l-V) = a*-h*. 3. Translate into symbols these verbal phrases and statements, using a and ^, or x and y, for the two numbers : (1) The sum of two numbers equals 25. (2) The difference of two numbers equals 15. (3) The sum of the squares of two numbers is less than 27. (4) The square of the sum of two numbers equals 100. /u USE OF LETTERS TO REPRESENT NUMBERS 355 (5) The difference of the squares of two numbers equals 9. (6) The sum of the squares of two numbers equals seven times the difference of the numbers. (7) The product of two numbers equals their sum. (8) The quotient of two numbers equals their difference. (9) A certain number increased by 1 equals another number diminished by 3. 4. Translate into words these symbolic expressions. For example, (1) means "one-ninth of the difference between C times a certain number and its square": (1)^; («)%-' = %"'; (11) y/mn=^ >/m \/n\ (2)^-;S (7)%i = ^t!; ^ y y + 1 (12)^-^^ = 1. y/x— y (3)^^ (8)^ = ».»; (13) a^V = (ai)«; (^)^; (») F -^ (1*)^ + 1 = (^)'+1' (^) .^2^ <-)? = (?)'' (15) y/a + l^y/a + 1. 5. Find the number whicli may be put in place of the letter in each of these equations to furnish true equations : (l)2a: + 3 = 7; Solution.— 2a? + 3 = 7 3 = 3 2x = 4 by Prindtple II, 2 2 a; = 2 by Principle IV. Check'. 207 + 3 = 2x2 + 3 = 4 + 3 = 7, which is correct. (2)3.r-l= 5; (4) ^7:+l = 2; («) ^+f = V; (3) 8a;-G = 18; (5) ^f -2 = 1; (7) -^ -1= V- Note.— First multiply both sides of (7) by 21. 6. i'ind the value of the numbers of problem 1 (8), (9), and (10). 3oG RATIOXAL GRAMMAR SCHOOL ARITHMETIC §202. Problems. — for either arithmetic or algebra. 1. The mercury column in a thermometer rose a certain num- ber of degrees one day, and 3 times as many degrees the next day. It rose 12° during the 2 days. How many degrees did it rise each day? Arithmetical Solution.— A certain number denotes the rise the first day. 3 times this number denotes the rise the second day. Hence 4 times a certain number denotes the rise in two days. 4 times a certain number equals 12° (by the given problem). Once the number equals 3°, the rise the first day (Principle IV). 3 times the number equals 9°, the rise the second day (Principle III). CJieck . 3° + ^° = 12°, the rise in two days. . Algebraic Solution.— Let X denote the first day's rise. Then, 3x denotes the second day's rise. X + 3x denotes the rise in 2 days. 4x = 12°. X = 3°, the first day's rise (Principle IV). 3x = 9^, the second day's rise (Principle III). Clieck: 3° + 9° = 12°. 2. A man bought 4 times as many hogs as cows, and after selling 5 hogs he had 23 hogs left. How many cows did he buy? Arithmetical Solution.—- A certain number represents the number of cows bought. 4 times this number represents the number of hogs bought. 4 times this number minus 5 denotes the number of hogs left. Then 4 times this number, minus 5, equals 23 (by the problem). 4 times this number equals 23 plus 5 (Principle I). 4 times this number = 28. This number = 7, the number of cows (Principle IV). Check: 4 X 7 — 5 = 23. Algebraic Solution.— Let X denote the number of cows bought. ! Then, 4x denotes the number of hogs bought. 4x — 5 denotes the number of hogs left. Then 4x — 5 = 23 (by the problem). 4x = 28 (Principle I). X = 7 (Principle IV). Ans. 7 cows. Check: 4 X 7 — 5 = 23. 3. Two masses were placed on one scale pan of a balance and found to weigh 18 lb. One of the masses was then placed in each pan, and it required 4 lb. additional on the light pan to balance the scales. What was the weight of each mass? USE OF LETTEBQ TO BBPBE8ENT NUMBERS 357 Arithmetical Solution.— A certain number of pounds denotes the weight of the heavier mass. Another number of pounds denotes the^weight of the lighter mass. The first number plus the second number denotes the combined weight (18 pounds). The first number minus the second denotes the difference of the weights, or the additional weight, which equals 4 pounds. 2 times the first number equals 18 plus 4 equals 22. The first number equals 11. 11 plus the second number = 18. (Principle IV). The second number = 7. (Principle II). The weights are, then, 7 lb. and 11 lb. Check: 11 lb. + 7 lb. = 18 lb. 11 lb. — 7 lb. = 4 lb. Algebraic Solution.— Let X denote the number of pounds in the weight of the heavier mass. Let y denote the number of pounds in the weight of the lighter mass. Then, x-^-y denotes the number of pounds in the combined weight, =1S X — y denotes the number of pounds in the additional weight, = 4. x + y = lS X — y = 4 2x = 22 (Principle I). X =11 (Principle IV). 11 + y = 18 2/ = 18 — 11 = 7 (Principle II). Check: 11 + 7 = 18 11 — 7= 4 Solve the following problems by both the algebraic and tho arithmetical method, and state which of the solutions is the shorter: 4. A man cut for me 3 times as many ash trees as oak trees, and as many hickory trees as ash trees and oak trees together. In all he cut 32 trees. How many of each kind did he cut? 5. A man bought 4 times as many 2^ stamps as 5^ stamps, and i as many 10^ stamps as 2^ stamps. For all he paid 14.95. How many of each kind did he buy? Note. — Let x denote the number of 5^ stamps purchased. 6. The side of a rectangular field is twice its breadth and the distance around the field is 240 rd. How many acres does the field contain? 7. I bought a number of books one day, 4 times as many the next day, and 3 books the third day. In all I bought 33 books. How many did I buy each day? 358 BATIONAL GRAMMAR SCHOOL ARITHMETIC 8. A newsboy sold a certain number of papers on Wednesday, twice as many on Thursday, and 3 more on Friday than on Thursday. In all he sold 48 papers. How many did he sell each day? Note.— Let x denote the number of papers sold on Wednesday. 9. Texas lacks 17,470 sq. mi. of being large enough to make 5 states the size of Illinois. How large is Illinois if Texas con- tains 265,780 square miles? 10. Maude has 78^, which lacks 18jJ of being 6 times as much money as James has. How many cents has James? 11. It is twice as far from Chicago to Niles by a certain route as from Niles to Albion, and from Chicago to Albion is 190 mi. How far is it from Chicago to Niles? 12. Rochester, N. Y., is 17 mi. more than twice as far from Chicago as is Detroit. Eochester is 587 mi. from Chicago. How far is Detroit from Chicago? 13. Niagara Falls is 8 mi. less than 9 times as far from Chicago, as is Michigan City. Niagara Falls is 514 mi. from Chicago. How far is it from Chicago to Michigan City? 14. It is 4 times as far, less 16 mi., from Chicago to N. Y. City as from Chicago to Ann Arbor. If it is 976 mi. from Chicago to N. Y. City, how far is it to Ann Arbor? 15. Champaign is 5 mi. more than ^ as far from Chicago as is Cairo. If it is 128 mi. from Chicago to Champaign, how far is it from Chicago to Cairo? 16. A father and his son together earn $75 a month, and the father earns 4 times as much as the son. How much does each earn? ^ 17. William has 3 times as many marbles as Joseph, and both together have 24. How many marbles has each? 18. A line 32 in. long is to be divided into 2 parts, one of which is 8 in. longer than the other. How long is each part? 19. A tree 75 ft. high was broken by the wind so that the part standing was 15 ft. longer than the part broken off. How long was each part? 20. The area of a rectangular field is 6400 sq. rd. and its length is 160 rd. ; what is its width? USE OF LETTERS TO REPRESENT NUMBERS 359 21. The area of a rectangular sidewalk is 120 sq. rd. and the width is 1^ yd. ; how long is the walk? 22. A man received $180 for 72 days' work; what was his daily wage? 23. Find the side of a square field equal in area to a rectangle 80 rd. long by 20 rd. wide. 24. A man received $3000 for 80 A. of land ; how much did he receive per acre? 25. The circumference of a circle is 2200 rd. ; what is the radius? (Equation: 6f r = 2200.) 26. The area of a circle is 201^^ m.*; how long is the radius? the circumference? 27. Find the radius and the circumference of a circle whose area is 1 square inch. §203. Formal Work. Find the value of the unknown quantity, x or y, in each of the following : 1. 6a; + 4 = 22. Note.— First multiply both mem- 2. 3^-10=14. bars of 12 by a.. ' 3. 4y-2.v + 2 = 16. 4. 8^4-^ + 3 = 21. 5. |a;-2 = 16. 6. |a;+5 = 35. Note. — First multiply both mem- bers by 3. 8.^^ = 2. 9. -g— - 5. 10. 6(a; + 2)=72. 11. !^=13. 12.^-15. X 13. 25 2a; = 5. 14. x-V 3a; +12 11 = 4. 15. 1x- -2a; + a; 5 = 18. 16. 21- -a; =33 -2a;. Note.— First add 2a5 to both ; bers, then subtract 31 from members. mem- both 17. 12a; +4= a; + 90. 4 18. 7rc-^ = 2a; + 39. o 19. .1^ . 3. 20. ?^^ = 7. 3G0 RATIONAL GRAMMAR SCHoOL ARITHMETIC 21. In the following equations W denotes the weight of the rope, or cable, in lb. per yd. ; Z the working load in tons; /S, the hreakinij load in tons, and 2), the diameter of the rope, or cable, in inches. For hemp cable, W^.b11D^\ Z = .109Z)^ >S'= .654Z>«. For tarred hemp cable, W^ 1.0362^^; L = .247i>2; S= 1.480i>^ For manila rope, W^ .7652^^ L = .329/)^; 8= 1.8772>^ For iron wire rope, W^ 3.847Z>'; L = 2.8622^^; S=11.0\2D\ For steel wire rope, W= 3.9462^^; L = 4.4412^^; 8^ 27.6302>^ Find the weight per yd., the working load and the breaking load of a hemp cable of 1" diameter; of 2^" diameter. 22. Solve similar problems for tarred hemp rope ; for manila rope. 23. Find TT, 2^, and 8 for an iron wire rope for which D = 1\"\ 24. For the same values of 2>, find PT, Z, and 8 for steel wire rope. §204. Equations Containing Two Unknown Numbers. In these problems two numbers will be unknown. The facts of each problem will furnish two equations. From these facts write both equations. Then change one or both equations, or combine them, in accordance with Principles I-V, pp. 350-2. PROBLEMS 1. The sum of two numbers is 10 and their difference is 4. Find the numbers. Solution. — Let x stand for the larger number. Let y stand for tlie smaller number. Then (1) x + y=10 (why?) and (2) x-y= 4 ('vhy?) Add Equations (1) and (2) member to member and write- (3) 2x =14. By which Principle? Then (4) X = 7. Which Principle enables us to get Equation (4) from Equation (3)? Putting this value, 7, of x, in place of x in Equation (1), we have (5) 7 + 2/ = 10 Subtracting 7 from both sides of Equation (5), we get (6) y= 3 EQUATIONS CONTAINIl^G TWO UNKNOWN NUMBERS 361 By which Principle do we have the right to subtract 7 from both sides of Equation (5) and then write the two differences equals The two desired numbers are then 7 and 3. N.B. — The j)roblem shows how to find any two numbers when their sum and their difference are known. 2. ThjB sum of two numbers is 18 and their difiference is 6. Find the numbers. 3. A piece of wire 8 ft. long is cut into two pieces, one of which is 2i ft. longer than the other. How long is each piece? 4. A 16-oz. mixture of acid and water contains 9 oz. more water than acid. What is the weight of each part of the mixture? 5. In a school of 40 children there are 12 more girls than boys. How many girls are in the school? How many boys? 6. The side of a rectangular lot is 75 ft. longer than the end and it is 350 ft. around the lot. Find the length, width and area of the lot. 7. The sum of two weights is 40 lb. and their difference is 8 lb. Find the weights. 8. I am thinking of two numbers. Their sum is 23 and their difference is 8. What are the numbers? 9. The top end of a May-pole 20 ft. high broke so that the part left standing was 5 ft. longer than the part broken off. How long was each piece? 10. A mixture of 18 oz. of vinegar and water contained twice as many ounces of vinegar as water. How many ounces of each liquid was in the mixture? Solution. — Equation (1) x + y = 18. Explain how this equation is obtained. Equation (2) x = 2y. How obtained? Write 2y in place of x in Equation (1), and have dy = 18. Find y. From Equation (2) then find x. Put the numbers found for x and y in Equation (1) in place of x and y and notice whether the result is atrue equation. This is called a' 'check. ' * 11. A mixture of water and acid weighing 28 oz. contains 3 times as much water as acid. How many ounces of each does the mixture contain? 12. A silver dollar weighs 412.5 grains and it contains 9 times as much pure silver as alloy. Find the weight of pure silver and of alloy in a silver dollar. 362 RATIONAL GRAMMAR SCHOOL ARITHMETIC 13 A piece of bronze weighing 15 lb. is made of copper and tin. It contains 4 times as much copper as tin. How mUch of each metal is there in the piece? 14. A bronze gun, weighing 25 tons, is made of copper and tin in the proportion of 9 times as much copper as tin. How much of each metal is there in the gun? 15. How many cubic inches of pure acid and of water are needed to make a gallon (231 cu. in.) of acid solution 12^% pure, i.e. containing 12^% of pure acid and 87^% of water? 16. The sum of one number and twice another is 11. The difference of the two numbers is 5. Find the numbers. 17. The sum of two numbers is 13. Twice the first number minus 3 times the second is 1. Find the numbers. 18. The sum of a number and half another number is 8^. Twice the first number minus 3 times the second is 5. Find the numbers. 19. Divide 24 in. into two parts, one of which is 6 in. greater than the other. 20. Divide 32 into two parts, one of which is 3 times the other. 21. Divide 28 into two parts, one of which is J of the other. 22. Two numbers are to each other as 3 to 5 and their sum is 64. Find the numbers. 23. If 1 be added to the numerator of a certain fraction it becomes i; but if 1 be added to the denominator, the fraction becomes J. Find the fraction. 24. Three times one number added to 2 times another number equals 23. Also 2 times the first number minus the second num- ber equals 6. Find the numbers. Solution. — ^The equations are Sx-\-2y = 23 and 2a; — 2/ = 6 Multiply both sides of the second equation by 2 and get 4x — 2y = 13. Now add this to the first equation and have 7x = 35. From this find x, then substitute the value of x in the second equation and find y. Check with the first equation. 25. Eight times a number minus another number equals 4. Three times the first number plus 5 times the second equals 66. Find the numbers. USES OF THE EQUATION 362 Suggestion. — Multiply the first equation through by 6 and add the second equation. 26. Two numbers are to each other as 5 to 7 and their sum is 6. Find the numbers. 27. Two numbers are as 2 to 7 and their difference is 25. Find the numbers. 28. One man said to another, "Give me one of your sheep and I'll then have as many as you." The other replied, "No; you give me one of yours and I'll have twice as many as you." How many sheep had each? Find the numbers the letters stand for in the following and check your work. r""^ . 33.| Sx - 2y = 20 30.i^ + r'« 34.j«^ + ^ = « (a:- 2^ = 8 \x -y = 3 \x + y = 15 I 6a + 26 = 17 _ i2x + dy=^d0 ^^ ( 3c + 6? = 21 ^^•|2:.-y = 10 ^^1^-2^ = 2 USES OF THE EaUATION §205. Thermometers. Fig. 250 shows the way the thermometer tube is marked off and numbered on the three thermometers most used by civilized countries. The Fahrenheit, or common, scale is on the left; the Centigrade scale, used everywhere in scientific work, is on the right; and the Reaumur (Ro'mur) scale, much used in Ger- many, is between the other two. On all scales the fundamental points are the freezing-jwint where the top of the mercury column stands when the bulb is in water containing melting ice, and the boiling-point, where the top of the mercury column stands when the bulb is in water commencing to boil. For any temperature the reading is the number which belongs to the mark on the scale at the top of the mercury column. 364 RATIONAL GRAMMAR SCHOOL ARITHMETIC REAUMUR rlOO AgphaUum nuUa Figure 350 1. What is the reading for "Water boils" on the Fahren- heit scale? on the Centigrade scale? on the Reaumur scale? 2. What are the three scale readings for "Water freezes"? 3. Into how many equal parts is the space between the boiling-point and the freezing- point divided on the Reaumur scale? on the Centigrade? on the Fahrenheit? 4. The lengths of those small spaces are marked off on the tubes above the boiling-point and below the freezing-point and numbered, as shown in Fig. 250. Notice how the numbers run on each of the scales between the freezing and the boiling points; below the freezing point. How should the numbering continue above the boiling point? 5. What do we call a change of temperature sufficient to expand, or contract, the mer- cury column an amount equal to one of the short spaces on the Fahrenheit scale? on the Centigrade scale? on the Reau- mur scale? These are written 1°P., 1°C., and 1°R., and are read "1 degree Fahrenheit," "1 degree Centigrade," and "1 degree Reaumur." USES OF THE EQUATION 365 C. If the readings above (f are written +1°, +2°, +3°, +5°, etCc , how should we write the readings below 0°? 7. What do the + and — then tell us about the readings? 8. Point out these readings in the figure: +2° F. ; +36°C.; +29°E.; -14°C.; -20°R.; -38°F.; -10°F. 9. Give the readings on each scale for the mean temperature at the Equator; at Rome; London; Chicago; Edinburgh; St. Petersburg; at the Poles. 10. Which degree is the longest, the 1°P., the 1°C., or the 1°R.? Which is the shortest? 11. How many Fahrenheit degrees are equal to 80 °R.? How many Centigrade degrees equal 180°F.? How many Reaumur degrees equal 100°C.? 12. On a sunny day find how many Fahrenheit degrees warmer it is in the sun than in the shade? Give the C and R readings corresponding to the two F readings and also the O and R differ- ences. §206. Applied Algebra (Graduation of Thermometers). The problems of §202, p. 356, show how the use of the equation simplifies some kinds of arith- metic problems. The work given here will illus- trate how some problems that would be quite, difficult by arithmetic can be easily handled by means of the equation. Experiments. — To convert thermometer readings from one scale to another: Three thermometer tubes, exactly alike, were filled with mercury to the same height. Their bulbs were immersed in a vessel containing melt- ing ice. Points were marked on the tubes at the tops of the mercury columns; beside the point on the first tube was written 32, and beside those on the second and third tubes was written. All three of the bulbs were then immersed in water commencing to boil, and the tops of the mercury columns were marked; beside the mark on the first 212 was written; beside that on the second, 100; and beside the mark on the third, 80. Let AB of Fig. 251. denote the straight line passing through the tops of the three mercury columns when the bulbs are in r If J j 1 <L F12-- ioo*- 3 X_ Y A_ _B 32' ^ 0' 1 1 1 1 F C R FIOI7BE 251 3G6 RATIONAL GRAMMAR SCHOOL ARITHMETIC melting ice. Let CD denote the line passing through the tops of these columns when the bulbs are in boiling water. Suppose the space lying between the line AB and the line CD on the first tube to be divided into 180 equal spaces and call each of these small spaces one degree Fahrenheit (written 1°F.). Denote its length (in in. or in cm.) by/. Suppose this same space on the second tube between the lines AB and CD to be divided into 100 equal spaces, and call one of these smaller spaces one degree Centigrade (written 1°C.). Denote its length (in in. or in cm.) by c. Divide the space on the third tube between the lines AB and CD into 80 equal spaces, and call one of these spaces one degree Reaumur (written 1°R.). Denote the length of this space (in in. or in cm.) by r. Notice carefully that no two of the lengths /, c, and r are equal. Denote the length of the space from AB io CD by 8. 1. 8 equals how many times/? c? r? 2. Since 180/ equals 8^ and 100 c equals 8^ how must 180/ compare with 100 c? (Answer this by writing an equation.) This problem exemplifies a new principle of much importance in using equations: Principle VI. — Numbers that are equal to the same number are equal to each other. 3. Since 180/ equals Sy and 80 r also equals 8, how must 180/ compare with 80 r? 4. If 180/ = 100c, /equals how many times c? 5. If 180/ = 80 r, / equals how many times r? 6. From these two equations, giving the value of /, write another equation by the aid of Principle VI. 7. Explain the meaning of the following equations: (l)/=|c; (2)/=|r, and(3)c = |r. Now suppose spaces equal to / marked off on the first tube above 212 and below 32, spaces equal to c marked off on the second tube above 100 and below 0, and spaces equal to r marked off on the third tube above 80 and below 0. The first tube is then grad- uated to the Fahrenheit scale, the second, to the Centigrade soale, and the third, to the Reaumur scale. USES OF THE EQUATION 3G7. §207. Equivalent Beadings on the Three Thermometers. On all thermometers, when the tops of the mercury columns are above 0, the readings are called positive^ and are marked +; when below zero they are called negative, and are marked -. If the thermometers are exposed to the same temperature between that of melting ice and of boiling water, the tops of the mercury columns will stand in a straight line, as XY. Let the mark that stands by this line at the top of the column on the first be called F; on the second top, C; and on the third, R. 1. Denoting by S the distance from the line AB to the line XF show by Fig. 251 that (1) 8=f (i^-32); (2) >S'= Cfc; and {3)8= Br. 2. Show by Principle VI that (4) / {F- 32)= cC; (5) / {F- 32) = rE; and (6) cC=rR, 3. Show from equations (1), (2), and (3), §206, problem 7, that (7) c= I/; (8) /' = |/ and (9) r= |6-, by using Principles III and IV. 4. In equation (4) problem 2 substitute c = |/ from (7) problem 3 and by Principles IV and II show that (I) F=^ ^Ch-32. 5. In a similar way show that the following equations are true ; (II) ^=172 + 32; (V) R = i {F-32); (III) ry= 1(^-32); {Vl)Ji = iC. {lY)C=iR; Note. — F, C, and R may be regarded as standing respectively for any • Fal ' • ~ • ' corresponding readings on the Fahrenheit, Centigrade, and Reaumur thermometers. §208. Problems. 1. Convert the following into their F. and R. equivalents by the aid of the proper equations (I) to (VI). MELTING TEMPERATURES ON CENTIGRADE SCALE. Ice 0.0° Zinc 412.0° Benzol + 4.4° Antimony 432.0" Tallow 43.0° Silver 1000.0° Paraffin 46.0° Copper 1100.0° Wax 62.0° Gold 1200.0° Sulphur 115.0° Cast iron 1200.0° Tin 230.0° Cast steel 1375.0° Bismuth 250.0° Wrought iron 1600.0° Cadmium 320.0° ?l»tinura 1775.0° Lead 826.0^ Iridium 1960.0* 368 RATIONAL GRAMMAR SCHOOL ARITHMBTIO 2. Convert these Fahrenheit boiling temperatures into their G. and R. equivalents : Ether +95° Carbon Bisulphide +114.8° Sulphuric Acid ... +14° Chloroform +141.8° Alcohol +172.4° Water +212° Mercury +674.6° Zinc + 1904° 3. Convert any observed Fahrenheit readings to their Centi- grade and Reaumur equivalent readings. 4. The heat of the body of a healthy man is 37.2°C. What are the Fahrenheit and Reaumur equivalents? 5. Solid carbonic acid dissolves in ether and the solution is a liquid of -90°C. What is the Fahrenheit equivalent? Solution.— In equation (I) if C = -90^ |C = -162". Then 162° meas- ured downward and 32° measured upward, or, -162^ *+ 32^= -130°. 6. For every vapor and gas there is a so-called critical temper- ature above which it remains in the gaseous condition, no matter how high the pressure upon it. The following are the critical temperatures of some vapors and gases: Ether vapor, +196°C. ; carbonic acid, +31° C; etheline, +9°C.; oxygen, — 118°C. ; nitro- gen, -145°C., and hydrogen, -174°C. What are the Fahrenheit equivalents? Laws of Thermometer Bepresented Graphically. When it is necessary to change a great many readings from one scale to another it is convenient to plot the equations. To illustrate this, take the equation ^= f C+- 32. Draw two perpendicular lines, as OX and OY, Fig. 252. Let all Centigrade readings be measured off horizontally and the corresponding Fahrenheit readings vertically to -^ any convenient scales. — The equation shows that if 50 CENTIGRADE RLA0INC5 C= 0°, F= 32° ; if C= 20°, F^ 68°; and so on. In this way we fill out FiGUBB 253 the following table : 8H0BTENIKG AKD CHECKING CALCULATIONS 3C9 0. F. C. F. 0° 32° 100° 212° 20° 68° 200° 392° 40° 104° . -20° -4° 50° 122° -40° -40° 75° 167° -80° -112° Note. — ^The numbers preceded by the sign — are readings below zero, and such numbers for C. must be measured off from toward the left, and for F. they must be measured off downward. Study the points on Fig. 253 and note whether they seem to lie on a straight line. With a ruler draw a straight line through these points. For minus (— ) values of C, as for C= —20, proceed thus: F=lx (—20) + 32 = 9 X (— ^ff°)-h 32 = 9 X (— 4) + 32 = —36 + 32. But 36° meas- ured downward and then 82° measured upward is the same , as 4° measured downward, or F= —4°, if 0= — 20"; and similarly for other minus ( — ) values of C. 1. Measure off the value = 60°, then measure vertically upward to the line drawn through the points, thus obtaining the corresponding Fahrenheit reading. What is F. for 0. = 60°? 2. Similarly, find from the drawing the values of F. corre- sponding to these values of C: 120°; 160°; 180°; 220°; -60°; -100°; 110°. 3. Make a similar table and plot of one or more of the other five equations of problem 5, p. 367. METHODS OF SHOBTENING AND CHECKING CALCULATIONS §210. Illustrations. 1. Additions, subtractions, multiplications, and divisions are conveniently checked by casting out the 9's, as explained on pp. 28, 54, and 75. 2. To check against gross errors (blunders), first think through the problem, making rough mental calculations with numbers that are approximately correct, and decide about what the result must be, 3. Check by performing reverse operations; that is, check addition by adding columns in reverse order ; check subtraction by adding the subtrahend to the remainder; check division oi square and cube roots by multiplication, and so forth. 370 RATIONAL GRAMMAR SCHOOL ARITHMETIC The second rule may be illustrated by a few problems: 1. Find the value of 15f A. of land at $87^. Think thus: 16 A. @ $90 would be worth $1440; at $85, 16 A. would be worth $80 less, or $1360. At $87i, 15^ A. of land is worth J of |85 less than 11440 — J of $80 (=$1400), or about $1383. The exact computation gives $1382.50. Or thus: 87i = I of 100. Therefore 15 J X 87i = 1580 x 5 = 7 X 197.5 = $1382.5. Aws. $1382.5. 2. On June 13, 1903, wheat is quoted in Chicago at 76f^ per bushel. Find the cost of 150 bu. at this price. 3. 250 shares of Illinois Central R. R. stock sold at $135^ a share. For how much did they sell? 4. The diameter of a circular rod is If". How many inches in the circumference pf a right section of the rod? How many square inches are there in the area of a right section of the rod? (For an approximation use w = 3i^.) 5. The outside diameter of a circular hollow iron tube is 2^" and the inside diameter is 2". How many cubic inches are there in a 12' length of the tube? 6. A distance was measured with a chain 98.75 ft. long and was found to contain 38.75 lengths of the chain. The chain was supposed to be 100' long. How great an error was made in measuring the line by using the supposed length? §211. Shortening and Checking Addition. 1. The noon temperatures on 7 successive days were 66**, 54**, 44°, 62°, 66°, 79°, 88°. Find the average for the week. 66 Mental Work. 70 + 50 + 44 = 164; 164 + 60 = 54 224; 224+2 = 226; 226 + 70 = 296; 296 — 4 = 292; 44 164 292 + 80 = 372; 372 — 1 = 371; 371+90 = 461; 461 — 226 62 2 = 459. 66 292 "^1^^^ ^^ called two-column addition. A little prac- 391 79 tice will make this method useful for 1, 2, or 3 columns g^ of figures. It may be used to advantage to check addi- — — tions'made in the ordinary way. 72459 Another method in use by expert accountants is to 65 6** Ans group the figures into sums of 10, 20, 30, and so on. Thus, in the given problem, 8 + 2. 6 + 4, 6 + 4, and 9 make 39. Then 7 + 3, 8 + 6 + 6, 6 + 4. 5, are 45. Sum, 459. 2. Write a few two and three column addition problems and practice these methods until you can use them rapidly. SHORTENING AND CHECKING CALCULATIONS ^Hl §212. Makingup Method of Subtraction. 1. A paying- teller in a bank had $5485 in his cash drawer in the morning, and during the day he paid out the following amounts: $37.50; $1G5.75; $10.25; $3.50; $2.88; $1.76; $05.17; $908.23; $3.67. How much money remained in the drawer? CONVENIENT FORM. MENTAL WORK.— Add the first column, Total, $5485.00 thinkiog thus: 10, 23, 31, 41. 41 and 9 make — zrzz the next larger number than 41 ending in ^^I'^l (the first figure in the total) Write the 9 in ^Jxlf *^^® result and add the 5 into the second col- ^^'l^ umn. Then 11, 21, 34, 48. 48 + 2 = 50, the ^•22 next number larger than 48 which ends in *-^2 (the second figure of the total). Write 2 in _1J2 ^1^® result and add 5 into the third column. J2nl Then, 21. 31, 39. 39 + 6 = 45. Write 6 and ^^•23 add 4 into fourth column. Then 10, 17, 26. 26 + ^'^" 2 = 28. Write 2 and add 2 to next column. $4226.29, balance Then, 12. 12 + 2 = 14, and finally, 1 + 4 = 5. Write the 4. The advantage of this method is that it foots all the numbers and subtracts their sum, at once, as the numbers stand in the account book, from the total, giving the balance directly. 2. A bank customer's deposit at the beginning of the month was $398.75. During the month he drew out the following amounts: $16.75^ $1.75; $5.25; $12.87; $1J!8.32; $40.45; $2.18; $9.16; $1.57; $11.38; $12.62. Find the customer's balance at the end of the month. §213. Shortened Multiplication. 1. Multiply 73 by 67. g^ = 7Q i 3 j 73 X 67 = (70 + 3) (70 -3) = 702 + ;|J x 3 - ? X ;r|J - 3^ = 702 _ 32 = 4900 — 9 = 4891. Algebraic form : (a + &) (a — 6) = a" — 61 This applies to finding the product of any two numbers the sum of whose units is 10 and whose tens digits differ by 1. Rule. — Find the difference bettueen the square of the tens and the square of the units in the larger number. 372 RATIONAL GRAMMAR SCHOOL ARITHMETIC 2. Find these products by the rule: (1) 34 X 26. (4) 58 x 42. (7) 71 x 69. (2) 22 X 18. (5) 65 x 55. (8) 99 x 81. (3) 43 x 37. (6) 98 x 82. (9) 95 x 85. 3. Find the square of 38. 38« = (30 + 8)2 = 302 + 2 X 8 X 30 + 82 = 1444 Algebraic form: (a + 6)2 = a2 + 2a6 + b*. Show the correctness of the following rule : Rule. — Square the tens, double the product of the tens hy the units and square the units, then add the three results. The sum is the square of the number. 4. Square these numbers by the rule : (1) 3G. (3) 64. (5) 19. (7) 125. (2) 47. (4) 58. (0) 95. (8) 148. Note.— Call the 12 in (7) 12 tens; also cjall the 14 in (8) 14 tens. 36 might be written 40 — 4. Then 362 = (40 — 4)2, (40 — 4)2 = 402 - 2 X 4 X 40 + 42. A Igebraic form : (a—hf = a'^ — %ab + 62. Make a rule for squaring 36 in this form. When long decimals are to be multiplied and the product is required to only a few decimal places, contracted multiplication is of great advantage. The next problem illustrates this sort of multiplication. 5. The radius of a circle is 238. 3G ft. What is the length of , the circumference to the second decimal place, or to the nearest 1 .01 foot? Note.— -The length of the diameter is 476.72 feet. Common Form 476.72 3.1416 Shortened Form 476.72 6141.3 digits reversed 28 47 1906 4767 143016 6032 672 88 2 1430.16 47.67 19.07 48 29 1497.66 3552 1497.67 SHORTENING AND CHECKING CALCULATIONS 373 Explanation. — ^In the shortened form the digits of the multiplier are written in reverse order, and the units digit is always written under that decimal place in the multiplicand which is to be the last one retained in the product. Multiply by units digit first, then by tens, and so on; in each case begin the multiplication by an^ digit with the digit just above it in the multiplicand. Begin the writing of each partial product in the same vertical line on the right. Note. — It is necessary on beginning to multiply by any digit ,to glance at the product by the preceding digit of the multiplicand to see how many units are to be added into the product by the digit just above. Thus, the multiplication by 4 would begin with 6, but 4 times the pr«* ceding digit (7) is 28, and this being nearly 8, the product 4x6 would be increased by 3, giving 27. Expert computers use the shortened form altogether. 6. Find the following products to the second decimal place by the method of shortened multiplication: (1) 36.428x3.1416. (3) 7.8843x1.0863. (2) 186.086 X 108.336. (4) 168.7431 x 28.329. 7. Find these products to .001 by shortened multiplication (1) 36.1872x6.8734. (3) 629.3865x3.1416. (2) 128.63x3.8629. (4) 1284.683x3.1416. §214. Shortened Division. 1. Divide 648.7863 by 68.372 to the nearest .01. Convenient Form 9.49— Quotient Explanation.— Find the units digit of 68.37)648.7863 the quotient in the usual way. Then cut off _^15_?2__ one digit from the right of the divisor and 33 45 find the next digit of the quotient, then cut 27 35 off another digit from the divisor, etc. A dot — — - is sometimes placed over each digit in the 6 10 divisor as it is set aside. 6 15 2. Find the following quotients to two decimal places: (1) 1786.786 -*. 3.1416. (2) 632.068 + 8.6249. (3) 1206.3862 + 28.3762. (4) 865.28476 4- 361.2946, 374 ItATlOKAL GRAMMAR SCHOOL ARITHMETIC §215. Shortened Square Eoot: Square Boot by Subtraction. 261 4 46 f 521 284 276 832 521 522)311.9(586 2670 449 417 32 Square root = 261.586 1. Find the square root of 68432.93628 correct to the third decimal figure. Explanation.— By placing a dot over every alternate digit in the given number, counting both ways from the decimal point, we obtain as many dots as there are to be figures in the square root. Find half the figures by the ordinary method of square root, and the rest by contracted division as shown. The result will very nearly equal the square root. 2. Find, by the shortened method, to two decimals the square roots of the following: (1) 6432. 18G4. (2) 38629.72468. (3) 8764.932651. 3 subtractions Square root may be found by subtraction, after noticing that 1 =1^ l + 3 = 2«; 1 + 3 + 5 = 3% 1 +3 + 5 + 7 = 4«; and so on. It is seen that the square root of the sum of the odd numbers in order from 1 upward is equal to the number of odd numbers added. An example will illustrate the use of this principle: 3. Extract the square root of 104976. . . . Explanation. — Place a dot over each 104976 ( 324=square root alternate digit beginning on the right, thus separating the number into groups of 2 digits each. From the first group on the left subtract 1, then 3, then 5, and so on as shown, until the remainder is less than the next odd number. The number of subtractions is the first root digit. In the present case it is 3. Bring down the next two-digit group. Double the root digit foimd and annex 1 to it, and subtract, then replace the 1 by 2 subtractions ^ *^d subtract, etc. The number of sub- tractions indicates the next root digit. Study the remaining steps and learn how to proceed further. This method gives the exact square root and may be used to check results found in the ordinary way. Some computing machines are based 647 on this method of obtaining the square 647 4 subtractions root. 4. Find the square roots, by subtraction, of the following: (1) 1156. (3) 174.24. (5) 54,756. (2) 44,944. (4) 49,284. (6) 289,444. 1 9 3 6 5 61 149 61 88 63 641 2576 641 1935 643 1292 645 SYNOPSIS OF DEFINITIONS PAGE 6. The average of two or more numbers is their sum divided by the number of them. (C/. pp. 85, 132.) 8. A board foot is a board one foot long, one foot wide, and not more than one inch thick. 10. Cash rent of farming land is a stated amount of cash per acre. Grain rent of fanning land is a stated part of all the crop. The tenant farmer is one who raises his crop on another man's fann. 12. Normal here means average. (Cf. p. 13.) The mean of two numbers is half their sum. 17. The digits (or figures) are the ten characters, 0, 1,2,3,4, 5,6,7,8,9. The name value of a digit is the value depending only upon the name of the digit. The place value of a digit is the value depending only upon the place of the digit. 22. Addition is combining numbers into a single number. The sum or amount is the result of the addition. The addends are the numbers to be combined or added. 26. The displacement is the number of tons of water a floating vessel pushes aside. 33. Subtraction means either of two things: (1) The way of finding the difference, or remainder, of two numbers. (2) The way of finding either one of two addends when their sum and the other addend are known. With the first meaning, the number from which we subtract is the minuend. The number to be subtracted is the subtrahend. The result is called the difference or remainder. With the second meaning, the known sum is the minuend. The known addend is the subtrahend. The unknown addend is the difference or remainder. 43. Literal numbers are those that are denoted by letters. 44. Multiplication of whole numbers is a short way of finding the sum of equal addends when the number of addends and one of them are given . The given addend is the multiplicand. The number of equal addends is the multiplier. The result or sum is the product. When a problem is expressed in an equation, the equation is called the statement of the problem. 375 376 BATIOXAL GRAMMAR SCHOOL ARITHMETIC The number which may stand in place of x in an equation is called the valv^ of x. A factor of a given whole number is one of two or more whole numbers, which, multiplied together, produce the given number. (C/. pp. 51, 142.) 52. A number which has factors other than itself and 1 is called a com- posite number. (C/. p. 143.) A niunber which has no factors other than itself and 1 is a prime number. (C/. p. 143.) 57. One inch of rainfall means one cubic inch of water for each square inch of horizontal exposed surface. The number of cubic units (cu. in., cu. ft., cu. yd., etc.) a vessel holds, when full, is called its capacity. 61. Division of whole numbers is a short way of subtracting one number from another a certain number of times in succession. (C/. p. 62.) 62. Division is a way of finding one of two numbers when their product and the other number are given. The product is called the dividend. The given number is the divisor. The required number is the quotient. We may say also that the dividend is the number to be divided, the divisor is the number by which the dividend is measured or divided, and the quotient is the measure. (C/. p. 61.) 78. A one-brick wall is a wall one brick thick, the bricks lying on the largest surfaces, the sides being exposed. 85. To average means to find the average. 86. The range of temperature is the difference between the highest and the lowest temperatures. 97. Such an expression as y = 10 is called an equation. The number on the left of the sign of equality is called the first mem- ber, or the left side, of the equation. The number on the right is called the second member, or the right side, of the equation. 98. The pencil point of the compasses is called the pencil-foot or the pen-foot. The other point is called the pin-foot. A circle is a curve such as is drawn with compasses. An arc of a circle is a part of a circle. The center of a circle is the point where the pin-foot of the compasses was placed to draw the circle. A diameter of a circle is a straight line joining two points of the circle and passing through the center. (C/. p. 107.) A radium of a circle is a straight line joining its center and a point of the curve. Circles whose centers are at the same point are called concentric circles. SYNOPSIS OF DEFINITIONS ' 377 101. Dividing a line into two equal parts is called bisecting the line. 102. An equilateral triangle is an equal-sided triangle. An isosceles triangle has at least two sides equal. (C/. p. 103.) That side of an isosceles triangle not equal to either of the two equal sides is called the base of the isosceles triangle. 103. A scalene triangle has no two sides equal. 104. A regular hexagon is a regular six-sided figure. 105. The sum of all the bounding lines of a figure is called the perimeter of the figure. Money is the common measure of the value of all articles that are bought and sold. 107. One of a whole number of equal parts of a given magnitude is called a fractional unit. (Cf. p. 138.) 109. A representation of an object, showing the various parts as folded back and spread out on a flat surface, is called a develop- ment. 110. A parallelogram is a quadrilateral whose opposite sides are parallel. (Cf. pp. 282, 283.) A rectangle is a parallelogram whose angles are (equal) right angles. (Cf. pp. 282, 283.) A square is both a rectangle and a rhombus. 111. The altitude of a parallelogram is the distance square across. The altitude of a triangle is the shortest distance to the base from the opposite comer. With all except isosceles triangles any side may be regarded as the base. 118. The mean solar day is the average time interval during which the rotation of the earth carries the meridian of a place eastward from the sun back around to the sun again. It is the average length of the interval from noon to the next noon. (Cf. p. 217.) 119. A section of land is a tract one mile square and containing 640 acres. 120. A township is a tract of land six miles square. 124. Per cent means hundredth or hundredths. 125. Interest is money paid for the use of money. (Cf. p. 252.) 132. The average rate of running is the distance divided by the time. 135. The ratio of one number (or quantity) to a second number (or quantity) is the quotient of the first number (or quantity) divided by the second. 136. The ratio of one magnitude to a second magnitude is called also the measure of the first by the second. The result of measuring one number by another is called the numerical measure of the first by the second. An equation of ratios is called a proportion. 138. The fractional unit of a fraction is one of the equal parts expressed by the fraction. (Cf. p. 107.) 378 RATIONAL GRAMMAR SCHOOL ARITHMETIC 139. The number above the fraction line is called the numerator (meaning numberer). The number below the fraction line is called the denominator (mean- ing namer). The numerator and the denominator are together called the terms of a fraction. 141. A fraction is said to be in its lowest terms when the numerator and the denominator are the smallest possible whole numbers without changing the value of the fraction The greatest common divisor (G. C. D.) of two numbers is their greatest exact common divisor. 142. Two numbers that have no common factor, except 1, are said to be prime to each other. A factor of a number is an exact divisor of the number, or a divisor that is contained without a remainder. (Cf. p. 44.) 143. A number that has no factors except itself and 1 is called a prime number. (Cf. p. 52.) A number that has factors besides itself and 1 is called a composite number. (C/. p. 52.) Any number that can be exactly divided by the number 2 is called an even number. All other whole numbers are called odd numbers. 147. A denominator which is common to two or more fractions is called a V common denominator, (Cf. p. 140.) When a common denominator is the least number that can be found which may be used as a common denominator of the fractions, it is called the least common denominator (L. C. D.). 148. A number that can be exactly divided by another nuitiber is called a multiple of the latter number. A number that can be exactly divided by two or more numbers is called a common multiple of those numbers. 149. The least common multiple (L. C. M.) of two or more numbers is the least whole number that is exactly divisible by each of the numbers. 151. A proper fraction is a fraction whose numerator is less than its denom- inator. An improper fraction is a fraction whose numerator is equal to, or greater than, its denominator. A mixed number is a number that is composed of an integer and a fraction. 160. To multiply a whole number by a fraction means to divide the multi- plicand into as many equal parts as there are units in the denom- inator, and to take as many of these equal parts as there are units in the numerator of the multiplier. 170. Dividing 1 by any number, whole or fractional, is called inverting the number. SYNOPSIS OF DEFINITIONS 379 The reciprocal of any number is the number inverted. 172. Fractions containing fractions in one or both terms are called complex fractions. The outside terms of a complex fraction (such as -y) are called the extremes, and the inside terms are called the means. 176. A straight line connecting the mid-point of a side of a triangle with the opposite comer is called a median of the triangle. The vertex of an angle is the comer. (The plural of vertex is vertices (v^r'-tX-ses.)) (C/. pp. 274, 288.) 178. To trisect a magnitude is to divide ijt into three equal parts. 179. A right-angled triangle is called a right triangle. 180. Parallel lines are lines running in the same direction. 182. A square is inscribed in a circle if the vertices of the square are all on the curve. 186. The first, second, third, and fourth numbers of a proportion are called the first, second, third, and fourth terms of the proportion. The first and the fourth terms of a proportion are called the extremes, and the second and the third terms, the means. The first two terms of a proportion are together called the first couplet; and the third and the fourth terms, the second couplet. 190. A dot, called the decimal point, or point, is used to show the units* digit. The point always stands (or is supposed to stand) just to the right of the units' digit or place. 191. The unit of the 1st place, or digit, to the right of the decimal point is called the tenth; of the 2d place, or digit, the hundredth; of the 3d, the thousandth; of the 4th, the ten-thousandth; and so on. 192. A decimal fraction, or decimal, is a fraction whose denominator is 10, 100, 1000, or some power of ten, in which the denominator is not written but is indicated by the position of the decimal p>oint. A power of 10 here means a number obtained by using 10 as a factor any whole number of times. 193. A pure decimal is a decimal whose value is less than one. A mixed decimal is a decimal whose value is greater than one. Numbers expressed in both decimals and common fractions are called complex decimals. A simple decimal is expressed without the use of common fractions. Finding the sum of decimal numbers is called addition of decimals. 195. Finding the difference of decimal numbers is called subtraction of decimals. 198. By the number of decimal places of a number is meant the number of digits (zero included) on the right of the decimal point. 204. The distance round a circle is sometimes called the circumference of the circle. (Cf. p. 107.) 380 RATIONAL GRAMMAR SCHOOL ARITHMETIC 207. The specific gravity of any solid or liquid substance is the ratio of its weight to the weight of an equal bulk of water. 209. Decimals that do not tenninate are called non-terminating decimals. Non-terminating decimals that repeat a digit or group of digits indefinitely are called repetends, or circulating decimals, or circulates. 211. A right section is a section made by cutting squarely across. 213. A denominate number is a number whose unit is concrete. A concrete unit is a unit having a specific name. A compound denominate number is a number expressed in two or more units of one kind. 215. A perch of stone is a square-cornered mass, l'Xli'Xl6i' = 24i cubic feet. A cord of firewood is a straight pile, 4' X 4' X 8'= 128 cubic feet. A cord foot is a straight pile of wood, 4'X4'Xl'= 16 cubic feet. 217. A leap year is a year of 366 days. A common year is a year of 365 days. 225. Reduction from higher to lower denominations is called reduction descending. Reduction from lower to higher denominations is called reduction ascending. 229. A m£ter is approximately one ten -millionth of the length of the part of a meridian of the earth, that reaches from the equator to the pole, called a quadrant of the earth's meridian. 231. An are is one square dckameter. A liter is one cubic decimeter. A gram is the weight of one cubic centimeter of distilled water at the temperature of its greatest density (39.1° Fahrenheit). 236. The number written before the sign "%" is called the rate per cent. The number, together with the sign "%," is called the rate. (Cf. p. 252.) 239. The result of finding a given per cent of any amount, or number, is called the percentage. The amount, or number, on which the percentage is computed is called the base. 243. Elevation means height above mean sea level. 248. Commission is a sum of money paid by a person or firm, called the principal, to an agent for the transaction of business. It is usually reckoned as some per cent of the amount of money received or expended for the principal. 249. A shipment of goods sent to an agent to be sold is called a consignment. A commercial (or a trade) discount is a certain rate per cent of re- duction from the listed prices of articles. The discount is usually allowed for cash payments or for payment within a specified time. (Cf. p. 260.) SYNOPSIS OF DEFINITIONS 381 252. Interest is money charged for the use of money. It is reckoned at a certain rate per cent of the sum borrowed for each year it is bor- rowed. (C/. p. 125.) When money earns 3, 6, 7, or 10 cents on the dollar annually (each year) the raie is said to be 3%, 6%, 7%, or 10% per annum (by the year), and the rate 'per cent is said to be 3, 6, 7, or 10. (C/. p. 236.) The sum of money on which interest is computed is called the principal. The principal plus the interest is called the amount. 259. A promissory note is a written promise, made by one person or party, called the maker ^ to pay another person or party, called the payee, a specified sum of money at a stated time. The sum of money for which the note is drawn is called the face value, or the face, of the note. The date on which the note falls due is called the date of maturity, and the time to run from any given date is the time yet to elapse before the note falls due. 260. Discount is a deduction from the amount due on a note at the date of maturity. (C/. p. 249.) The sum of money which, at the specified rate and in the time the note is to run before falling due, will, with interest, amount to the value of the note when due, is called the present worth of the note. The difference between the value of the note, when due, and the present worth is called the true discount. The bank discount of a note is the interest upon the value of the note when due, from the date of discount until the date of maturity. 261. When a note or bond is paid in part the fact is acknowledged by the holder by his writing the date of payment, the sum paid, and his signature on the back of the note or bond. This is called an in- dorsement. 264. The small wheels under the front of a locomotive engine are called engine truck wheels, or leaders. The large wheels are called drivers. The smaller wheels just behind the drivers are called trailers. 266. Tractive force is pulling (or drawing) force. 267. Replacing a letter (in an equation) by a number is called substituting the number for the letter. Performing the operations indicated in an equation and obtaining the number-value of a letter is called finding the value of that letter. 274. The surfaces of the cube meet each other in edges, thus forming lines. The corners are called vertices. A single comer is a vertex. (Cf. pp. 176, 288.) The edges meet each other in comers of the cube, thus forming points. 278. When two lines meet, making the angles at their point of meeting (intersection) equal, the lines are said to be perpendicular to each other, and each is called a perpendicular to the other. 382 RATIONAL GRAMMAR SCHOOL ARITHMETIC The angles thus formed are called right angles. 279. Lines which go through the same point are called concurrent lines. 282, 283. A quadrilateral is a four-sided figure. 282, 283. A rhombus is a parallelogram whose sides are equal. 283. A rhomboid is a parallelogram that is not a rectangle. A trapezoid is a quadrilateral having at least one pair cf opposite sides parallel. 285 A quadrant of arc is one quarter of a complete circle. An angle is the amount of turning of a line about a point as a pivot. It may also be regarded as the difference of direction of two lines. A straight angle is the sum of two right angles. 286. An angular degree is one of the ninety equal parts of a right angle. A degree of arc is one of the ninety equal parts of a quadrant. A sextant is one-sixth of a complete circle. An octant is one-eighth of a complete circle. 287. A perigon is an angle equal to one complete revolution. 288. If two lines intersect each other, two angles lying opposite to each other are called opposite or vertical angles. The lines which include an angle are called the sides of the angle The point where the sides meet is called the vertex of the angle. (C/. pp. 176, 274.) 289. An angle that is smaller than a right angle is called an acute angle. An angle that is larger than a right angle and less than a straight angle is called an obtuse angle. 291. Two angles whose sum equals a right angle, or 90®, are called com- plemental angles. Two angles whose sum equals two right angles, or 180°, are called supplemental angles. 297. An imaginary curved line, called the equaior^ divides the surface of the earth into the northern and the southern hemispheres. 298. The imaginary curved lines running east and west are called parallels; those runnmg north and south are called meridians. {Cf. pp. 299, 310). 299. Longitude is the distance in degrees, minutes, and seconds (of arc) measured on a parallel due eastward or westward from a chosen meridian called the prime meridian. {Cf. pp. 297, 300.) Latitude is a similar distance measured northward or southward on a meridian. 305. The date line is the 180th meridian. 306. A four-sided figure (a quadrilateral) having at least one pair of parallel sides is called a trapezoid. The altitude of a trapezoid is the distance square across between two parallel sides called the bases of the trapezoid. 307. The lengths of the bases and of the altitude of a trapezoid are called its dimensions. SYNOPSIS OF DEFINITIONS 383 A square of roofing is a ten -foot square (100 sq. ft.). Shingles are said to be laid so many inches to the weather vrhen the lower end of each course of shingles extends so many inches below the course next above it. 310. A toiVnship is a tract of land six miles square. A range is a tier (or row) of townships running north and south. 311. A section is a tract of land one mile square. Such of the north and the west rows of half-sections of a township as do not have exactly 320 acres each are called lots. 312. The volume of any figure is the number of cubical units within its bounding surfaces. * 317. A straight line connecting two points of an arc is called a chord of the arc. 318. A circle is said to be circumscribed around a triangle and the triangle is said to be inscribed in the circle if all the vertices of the triangle are on the curve. 319. The longest side of a right triangle, that is, the side opposite the right angle, is called the hypothenuse, 321. ^The product obtained by using any number twice as a factor is called the square of that number. 322. The square root of a number is one of its two equal factors 326. The cube of a number is the product obtained by using the niunber three times as a factor. 327. The cube root of a number is one of its three equal factors. 328. In triangles having the same shape, angles lying opposite pro- p>ortional sides in different triangles are called corresponding angles. In triangles having the same shape, sides lying opposite equal angles are called corresponding sides. 336. The amount paid for insurance is called the premium. The written agreement between an insurance company and the insured is called a policy. The amount for which the property is insured is < aVed the Jace of the policy. 338. A tax is a sum of money levied by the proper officers to defray the expenses of national, state, county, and city governments, and for public schools and public improvements. Assessed valuation means the estimated value of the property that is assessed. 341. A stock (called also a stock certificate) is a written agreement made by a company to pay the holder a certain part of the earnings of the company. When a stock company pays to the holders of its stock $2 on each $100 of its capital stock, or 2% on its stock, the company is said to be paying a %2 dividend, or a 2% dividend. 384 RATIONAL GRAMMAR SCHOOL ARITHMETIC At par means at its face value. Above par means at more than its face value. Below par means at less than its face valu^. A bond is a written agreement made by a national, state, county, or city government, or by a company, to pay the holder interest at a stated rate on a stated sum of money, called the face of the bond. A broker is a man that makes a business of buying and selling stocks and bonds for other people. A broker's charge for his services is called brokerage. 342. Preferred stocks are stocks which pay a fixed dividend before any dividends are paid on common stocks. Dividends are paid upon common stocks after expenses and dividends on preferred stock have been paid Government 2s are government bonds paying 2% interest. 343. Compound interest is interest on interest. The compound amount is the principal plus the compound interest. 345. Coupon notes are notes, attached to interest-bearing notes, for the amount of interest due each at interest-paying period. 349. An expression in :which the sign < or > stands between two numbers is called an expression of inequality. 350. The signs =, < , and > are called relation signs. General Defilnitions Quantity is limited magnitude. Number is the result of the measurement of quantity. Number may also be defined as the ratio of quantities of the same kind. Arithmetic is the science of numbers and the art of using them. GENERAL INDEX Explanatory. — The following index has been prepared for daily use by young students of^^various ages, and the principal purpose underlying its construction has been to enable all studfents to find readily and quickly the thing desired, that it may be in use not occasionally but continually. To this end the entries have been arranged not merely for the noun (or the principal noun) but also, in many cases, for other words of the phrase^ thus making the index usable also by students to whom grammatical distinctions are not entirely familiar. In reading, the phrase after the comma (where one occurs) is often to be read first; thus "addition, short methods for" is read "short methods for addition." The references are to pages. Where two page references are sepc,rated by a dash the meaning is "and included pages;" thus 27-30 means pages 27, 28, 29, 30. Angle and of arc, minute of, 287, 289 Angle and of arc, second of, 287, 289 Angle, sides of, 288 Angle, vertex of, 176, 288 Angles, addition of, 289-293 Angles and arcs, measuring, 285-289 Angles (and figure), opposite or verti- cal, 288 Angles, complemental, 291 Angles, difference and sum of, 289-293 Angles, division of, 176 Angles formed by parallel lines and a line cutting them. 288 Angles of hexagon, sum of, 292 Angles of n-gon, sum of, 293 Angles of octagon, sum of, 293 Angles of parallelogram, sum of, 292 Angles of quadrilateral, sum of, 292 Angles of right triangle, sum of acute, 292 Angles of similar triangles, correspond- ing, 328 Angles of triangle, sum of. 291 Angles, subtraction of, 290 Angles, sum and difference of, 289-293 iUigles, supplemental, 291 Angular degree, 286, 289 Angus cattle, weights and prices of, 203 Annually, 252 Apothecaries' weight, 214 Applications of percentage, 336-345 Applications of proportion, practical, 188-190, 327-336 Applications to transportation problems, see also tractive force, 264-271 Arabic notation, 20 Arabic numeral, 20, 21 Arc, 98, 285 Above par, 341 Account, forms of, 5, 47-49, 88-91 Accounts, bills and, 88-96 Accounts, farm, 47-50 Accounts rendered, 91 Acquired territory of United States, area and dates of, 246 Acre, 215 Acres covered by one mile of furrow of given width, number of, 161 Acute angle (and figure), 289 Addend, 22 Addition, checking, 24, 28, 369, 370 Addition of angles, 289-293 Addition (of common numbers), 21-32 Addition of decimals, see also decimals, 193-195 Addition of fractions, see fractions Addition of fractions having common fractional unit, 146 Addlfion, short methods for, 369, 370 Admission of states, territories, etc., dates of, 39 Air, pressure oL 127, 128 Almanac, 244, 245 Altitude, see also elevation and height Altitude of parallelogram. 111 Altitude of trapezoid, 306 Altitude of triangle. 111 Amount, 22, 252 Amount, compound, 343 *'And" In decimals, use of, 191 Angles, 285 Angle (and figure), acute. 289 Angle (and figure), obtuse, 289 Angle (and figure), right, 278, 289 Angle (and figure), straight, 285 An^Ie and of arc, degree of, 286, 289 885 386 RATIONAL GRAMMAR SCHOOL ARITHMETIC Arc, degree of angle and of, 286, 289 Arc, minute of angle and of, 287, 289 Arc, second of angle and of, 287, 289 Arcs, measuring angles and, 285-289 Are (unit of land measure), 231 Area, 31, see also measuring surface and mensuration Area and population of States and Ter- ritories, 30, 35 Areas of common forms, 132-134 Areas of countries, 35, 09. 70 Areas of fields, 9, 10 Areas of river basins, 37 Ascending, reduction, 225 Assessed valuation. 338 Average, 6, 85. 132 Avoirdupois weight, 214, 220 Bale {unit of paper measure), 217 Bale of cotton, weiglit of, 161 Ball, velocity of cannon, 83, 206 Ball, velocity of rifle, 206 Bank discount, 260 Barrel, number of gallons in, 34, 228 Barrel, number of pecks In, 71 Base (in percentage), 239 Base line, 310 Base of isosceles triangle, 102 Base of parallelogram. 111 Baseball diamond (and figure), 2, 3 Bases of trapezoid, 306 Basins, areas of river, 37, 247 Battleships, data of, 26 Beans, capacity for absorbing water, 195, 196 Beef, food parts of, 190 Below par. 341 Billion, 18, and footnote Bills and accounts, 88-96 Birds, speeds of, 75, 206 Bisect, 101, 176 Bisector, figure of perpendicular, 101 Bisector, perpendicular, 176 Block and lots (and figure), town, 76 Blocks, figure of city streets and, 306 Board foot, 8, 227 Board measure, see measuring wood, 8 Body, average surface of human. 128 Body, dailv requirement of water and of solid food for human, 145 Body, normal temperature of a man*s, 368 Boiling point. 116, 363 Boiling temperatures, 364, 368 Bond, face of, 341 Bond quotations and transactions, stock and. 341 Bonds, stocks and. 341-343 Book, figure of, 184 Box, capacity of square-cornered, 57, 58, 112 Breeze, kinds of. 14 Brick wall, kinds of, 78, footnote Brick work. 78-80, 308, 309 Broker. 341 Brokerage. 341 Bulk, see volume Bundle (unit of paper measure), 217 Bnshel, 210 Bushel, number of cubic feet of grain In, 112, 205 Bushel, number of cubic inches in, 113 Bushel of various articles, weights of, 71, 114, 216 Bushel, Winchester, 216 Butcher's price list, 4 Cables, data concerning, 360 Calculations, methods of shortening and checking. 51-53, 60, 68, 70-75, 191. 192, 369-374 Calendar, Gregorian, 217 Calendar, Julian, 217 Calendar month, 217 Calendar year, 217 Cancellation, 72, 82, 83. 158, and note Cannon ball, velocity of, 83, 200 Capacity. 57 Capacity, measuring, 112, 113, 210. 231 Carat, 220 Carrier pigeon, speed of, 206 Cars, engines and tenders, heights, lengths and weights of railroad. 15, 64, 129. 130, 264 Cash rent, 10 Casting out the nines, 28, 53, 54, 75, 369 Cattle at Chicago Stock Yards, receipts of, 35, 36 Cattle, weights and prices of Angus, 203 Cattle weights and values of, 50 Cent, 213 Centare, 231 Center of circle, 98 Center of square, 320 Centeslmi, 214 Centi-, 2.30 Centigrade thermometer, 303-309 Centime, 214 Central time, 303 Centre. See center Century, 217 Certificate, stock. 341 Chain, also square chain, 215 Chair, figure of. 184 Change of notation, 21 Checking calculations, methods of short- ening and, 369-374 Checking calculations : ' Addition, 24. 28, 369, 370 Casting out the nines, 28, 53, 54, 75, 369 Division. 68. 74. 75. 369. 373 Multlnlication. 53, 54, 309, 371-373 Square root, 374 Subtraction. .S69, 371 Cheat expansion. 106 Chest measure. 12 Chicago Stock Yards, receipts of cattle at. 35. .SO Chord (of a circle), 317 Circle (and figure). See also ctrcum- ference-on<t mensuration. 98, 289 Circle and parallelogram compared (and figures). 210 Circle, area of, 210, 211 Circle, center of, 98 Circle, hour, 804 ^ GENERAL INDEX 387 Circles, concentric, 98 Circulate, 209 Circulating decimal, 209 Circumference. See also circle and men- suration, 107, 204, 205, 289 Circumference to diameter, ratio of, 204, 205 Circumferences of various tilings. See also diameters, etc., 56, 106-108, 159, 2D4, 205, 211 Circumscribed, 318 Cities, populations of, 28, 41 City streets and blocks, figure of, 306 Clock, figure of, 285 Coal, hard vs. soft, 4 Coal in ton, number of cubic feet of hay and of, 205 Coast lines of continents, lengths of, 246 Coffee Imports of United States, 27, 56, 203 Coins, fineness of United States gold and silver, 55, 226, 361 Coins, weights of United States, 50, 55, 155, 156, 194, 195, 226 Commerce. 38-41 Commission, 248, 249 Common denominator, 140, 147 Common fraction. See fractions Common fraction, reduction of decimal to, 192, 193 Common fraction to decimal, reduction of. 208, 209 Common multiple, 148 Common stock, 342 Common uses of numbers, 127-134 Common year, 217 Comparison of prisms, 276-278 Compasses (and figure), pen-foot and pin-foot of, 98 Complemental angles, 291 Complex decimal, 193 Complex fractions, 172, 173 Complbsite factors, 142, 143 Composite number, 52, 143 Compound amount, 343 Compound denominate numbers, 213-235 Compound interest, 343-345 Concentric circles, 98 Concrete unit, 213 Concurrent lines, 279-281 Cone, model and development of right circular, 314 Cone, volume of right circular, 314 Consignment, 249 Constructions. {See also model and de- velopment) : Drawings: I. Circle with given radius, 98 II. Line equal to given line, 98, 99 III. Line equal to sum of two or more given lines, 99 • IV. Line equal to difference of two given lines, 100 V. Line equal to two. three or four times given line, 100 VI. and I. Divide given line Into two equal parts (bisect It), 100, 101, 176 VII. Equilateral triangle with each side equal to given line, 102 VIII. Isosceles triangle with sides equal to two given lines, 102 IX. Scalene triangle with sides equal to three given lines, 103 X. Three-lobed figure in circle of given radius, 103, 104 XI. Regular hexagon in circle of given radius, 104 I. Divide given line Into two equal parts (bisect it), 176 II. Divide given angle into two equal parts (bisect it), 176 I. Line parallel to given line, 177 II. Line parallel to given line and through given point, 178 III. Line parallel to given line and through given point, 178 IV. Divide given line Into three equal parts (trisect it), 178 V. 30** and 60 ^ and 45% right triangles, 180 VI. Through given point, line paral- lel to given line, 181 VII. Through given point on given line, perpendicular to line. 182 VIII. Through given point out of given line, perpendicular to line, 182 I. Perpendicular to given line from given point on It, 272 II. Perpendicular to given line from given point out of it, 272, 273 III. Square in circle of given radius, 273 Greasings (paper-folding) : I. Perpendicular to given line, through given point on It, 278 II. Bisect an angle, 279 III. Three non-parallel lines, 279 IV. Bisect three angles of triangle, 279 V. Bisect given line. 280 VI. Square and Its diagonals, 280 VII. Three perpendiculars from vertices of triangle to opposite sides, 281 VIII. Three perpendicular bisectors of sides of triangle, 281 IX. Rectangle and Its diagonals, 281 Cuttinos: Find sum of two given angles. 290 Find difference of two given angles, 290 Find sum of angles of equilateral triangle, 291 Find sum of angles of scalene tri- angle, 291 Find sum of angles of right trian- gle, 292 Find sum of acute angles of right triangle, 292 Find sum of angles of quadrilateral. Find sum of angles of hexagon. 292 Find sum of angles of octagon. 293 Find sum of angles of n-gon, 293 388 RATIONAL GRAMMAR SCHOOL ARITHMETIC Find relations of parts of rectangle with a diagonal, 293 Find relations of parts of parallel- ogram with a diagonal, 2d3 Dratoings: I. Center of given arc, 317, 318 II. Bisect given arc, 318 III. Circumscribe circle about equi- lateral triangle, 318 IV. Trefoil, 318 V. Quatrefoil, 318 VI. Designs of figure 216, p. 319 VII. Sixfoil, 319 VIII. Right triangle, 319 IX. Right triangle having given hypothenuse, 319 X. Right triangle having given two legs, 320 Cuttings: XI. Find relation of squares on sides of isosceles right triangle, 320 XII. Find relation of squares on sides of any right triangle, 320. Constructive geometry. See also con- structions, 98-105, 176-182, 272- 281, 289-293, 307-320 Copeck, 214 Cord {unit of fire-wood measure), 215 Cord foot. 215 Corresponding angles of similar trian- gles. 328 Corresponding sides of similar triangles, 328 Cost of living, 3-6, 93-96 Cotton, weight of bale, 161 Counting, 217 Counting, measurement by, 217 Countries, areas and populations of, 70 Couplets, first and second, 186 Coupon notes, 345 Creasing paper. See a2«o . constructions, 278-281 Critical temperatures of gases, 368 Crow, speed of, 75, 206 Crown, 213, 214 Cuba, area of, 35 Cuban school data. 26 Cube, model and development, of three- inch, 274 Cube of number. 326 Cube root of number. 327 Cube roots, cubes and, 326, 327 Cubes and cube roots, 326, 327 Cubes of units and tens, table of. 326 Cubic feet of hay and coal In ton, num- ber of, 205 Cubic foot, number of gallons In, 266 Cubic foot of various things, weights of. 56, 64. 82, 114, 161, 165, 207, 208, 235, 266 Cubic foot of water, weight of, 165. 235, 266 Cubic Inch of steel, weight of. 113 Cubic Inch of water, weight of, 164 Curves, plotted, 86, 120. 122. 267, 368 Cylinder, lateral surface of right circu- lar, 313 Cylinder, model and development of right circular, 313 Cylinder of engine, figrnre of, 211 Cylinder, volume of ri£^t circaUr, SU Dairying, 16, 50, 75, 204. See aUo miik ana mlikings Date line, 305 Date of maturity, 259 Dates of acquisition of territory o! United States, 246 Dates of admission and population of states, territories, etc., 39 Day, 21/ Day, mean solar, 118, 217 Decl-, 230 Decimal, 192 Decimal, circulating. 209 Decimal, complex, 193 Decimal fraction. See decimal, 192 Decimal, mixed, 193 Decimal, non-terminating. 209 Decimal notation, 19, 190. 191 Decimal places, number of, 198 Decimal point, 190 Decimal point in product, 198, 199 Decimal, pure. 193 Decimal, reduction of common fraction to, 208, 209 Decimal, simple, 193 Decimal to common fraction, reductloa of, 192, 193 Decimals, 190-212 : Notation, 190, 191 Numeration, 191, 192 Reduction of decimal to common frac- tion, 192, 193 Addition, 193-195 Subtraction, 195-197 Multiplication ( pointing off the prod- uct), 198. 199 Division, 200-208 Of decimal by integer, 200-202 By decimal, 200-208 Reduction of common fraction to deci- mal, 208. 209 Decimals, addition of, 193, 194 Decimals, division of. See decimals, 200-203 Decimals, notation of, 190, 191 Decimals, numeration of, 191, 192 Decimals, principles of. See principles, etc. Decimals, subtraction of 195-197 Degree, angular, 286, 289 Degree of angle, 286, 289 Degree of arc, 286, 289 Degree of longitude at equator, length of, 164 Deka-. 230 Denominate number, 213 Denominate numbers, compound, 213- 235 Denominator, 139 Denominator," common, 140, 147 Denominator, least common, 147 Depth, greatest ocean, 69 Depths of lakes, 248 Descending, reduction. 225 Description of land, 310-312 Development, 109, 275 Developments. For list see model, et€i. GENERAL INDEX 389 diagonal, 179/ 292 :>iagonai divides figure, how, 179, 292, 293 Diameter, 98, 107 Diameter, ratio of circumference to, 204, 205 Diameters of eartli, 206, 315, 317 Diameters of heavenly bodies, 317 Diameters of various things. See also circumferences, etc., 107, 108, 204, 205, 211 Diamond (and figure), baseball, 2, 3 ■Difference, 33 Difference of angles, sum and. 289-293 Differences of lines, products of sums and, 293-295 Digit or figure, 17 Digit, name value of, 17 Digit, place value of, 17 Dime, 213 Dimensions of trapezoid, 307 Discpunt, 249, 260 Discount, bank, 260 Discount, trade, 249, 250, 339, 340 Discount, true, 260 Discounting notes, 260, 261 Displacement of vessel, 26 Distance, measuring, 106-108, 214, 215 Distances from Chicago to various cities, 45, 55, 128, 206, 358 Distances from Washington to various cities. 54, 55 Distribution of population of United States, 1900. 29-31 Dividend, 62, 341 Divisibility of numbers, 73, 74. See also factors, prime and composite, and greatest common divisor Division and multiplication compared, 61-63 Division and subtraction compared, 60, 61 Division by multiples of 10, 70-72 Division, checking, 66, 68, 74, 75, 369, 373 Division, long, 65-70 Division (of common numbers), 60-87 Division of decimals. See decimals Division of fractions. See fractions Division of lines and angles, 176 Division, short, 03. 64 Division, short methods for, 70-74, 191, 373 Divisor, 62 Divisor, greatest common, 141, 143 Dollar, 105, 213 Door, figure of, 183 Double ea^le, 213 Dozen, 217 Dram, 214 Drivers of locomotive 56, 264, 270 Dry gallon, liquid and, 216 Dry measure, 216 Duck, speed of wild, 206 Eagle ; half-eagle ; quarter-eagle ; double- eagle. 213 Earth, area of : land area of ; water area of, 246 Earth, diameters of, 206, 315, 317 Earth, mean diameter of, 315, 317 Earth, mean radius of, 19, 316 Earth, velocity of, 83 Eastern time, 303 Electricity, velocity of, 206 Elevation. See also altitude and height Elevations and heights of mountains, 37, 69, 243, 247 Elevations of lakes, 35, 37, 248 Elevations of Weather Bureau stations, 243 Engine, figure of, cylinder of, 211 Engines and tenders, heights, lengths and weights of railroad cars, 15, 64, 129, 130, 264, 266 English notation, 18, and note Equal parts, fractions as ratio and as, 138-140 Equation, 44, 97 Equation, first and second members of, 97 Equation, left and right sides of, 97 Equation, principles for using. See also principles, etc., 97, 350-352, 366 Equation, uses of, 348, 360-369 Equations with two unknown numbers, 360-363 Equator, 297 Equilateral triangle (and figure), 102, 103, 282, 284, 326 Equivalent readings of thermometers, 367-369 Equivalents, metric and United States, 230-232 Equivalents of longitude and time, 305 Even number, 143 Expansion, chest, 106 Expense account, family, 93-96 Expenses of living, 3-6, 93-96 Exponent, 143 Export trade, 1891. 1901, United States, 40 Express trains, speed of, 82. 83 Expression of Inequality, 349 Extremes, 172, 186, 187 Pace (of a note), 259 Face of bond, 341 Face of policy, 337 Face value, 259 Factor, 44, 51, 142 Factor, prime, 142, 143 Factors, greatest common divisor by prime, 143, 144 Factors, multiplication by, 51, 52 Factors, order of, 160 Factors, prime and composite. See also divisibility of numbers, 142, 143 Factory life upon growth, effect of, 197 Fahrenheit, thermometer, 116, 363-369 Falcon, speed of. 206 Family expense account, 93-96 Farm accounts, 47-50 Farm, fencing, 7-9, 133 Farm, figure of, 7 Farm products of United States, 83-85 Farthing, 213 Fence wire, cost and weight of, 7, 8, 133, 1,S4 Fencing farm, 7-9, 133 Fields, areas of, 9, 10 390 RATIONAL GRAMMAR SCHOOL ARITHMETIC Figure or digit, 17 Filler, 213, 214 Finding the value of (phrase), 267 Fineness of United States gold and sil- ver coins, 55, 226 Fire-wood by cord, measuring, 215 Flat prism, model and development of, 276 Folding paper. Bee also constructions, 278-281 Food parts of beef, 190 Food, prices of. 4 Foot, board, 8, 227 Foot, cord, 215 Foot, number of gallons in one cubic, 266 Foot; aXao square and cubic foot, 214, 215 Foot of compasses, pen-foot and pin- foot, 98 Force. See tractive force Forces, Joint effect of, 173-175 Forms, areas of irregular but common, 132-134 Forms, figures of irregular but common, 132-134, 284 Forms of account, 5, 47-49, 88-91 Formula, 239 Foundation walls, 79 Fraction as ratio and as equal parts, 138-140 Fraction, common, 192. Bee also Frac- tions Fraction, decimal, 192. Bee aUo deci- mals Fraction, Improper, 151 Fraction, lowest terms of, 141-142 Fraction, proper, 151 Fraction, reduction of decimal to com- mon, 192. 193 Fraction, terms of, 139 Fraction to decimal, reduction of com- mon, 208, 209 Fractional numbers, multiplication by, 54 Fractional unit. 107, 138 Fractions, 135-100 Introduction (ratio and proportion), 1,35-140 Ratio (measure), 1.35, 136 Proportion, i:i«. 137 Fractions ns ratios and as equal parts. 138-140 Fractions (simple), 140-172 Reduction to higher, lower and low- est terms, 140-142 Factors, prime and composite, 142, 143 Greatest common divisor by prime factors, 143. 144 Addition of frnctions having com- mon fractional unit, 146 Addition of fractions easily reduced to common fractional unit, 146, 147 Multiples and least common multi- ple, 148-151 Reduction of whole and mixed num- bers to fractions, and inversely, 151-153 Addition. 153-155 Subtraction, 155-157 * Multiplication, 157-165 Of fraction by whole namber, 157-158 Of mixed number by Whole num- ber, 159 Of whole number by fraction, 160, 161 Of fraction by fraction, 162, 163 Of mixed number by mixed num- ber. 163-165 Division, 165-172 Of fraction by whole nnmber, 165- 167 Of mixed number by whole num- ber, 167, 168 Of any number by fraction, 168- 172 Fractions, addition of. Bee also frac- tions, 153-155 Fractions, complex, 172, 173 Fractions easily reduced to, common fractional unit, 146, 147 Fractions having common fractional unit, 146 Fractions, Introduction to, 135-140 Fractions, principles of. Bee principles, etc. Fractions, reduction of. Bee also deci- mals and fractions, 140-142, 146, 147. 151-153, 192, 193, 208, 209 Fractiotas, subtraction of. Bee fractions Fractions to higher, lower and lowest terms, reduction of. 140-142 Fractions with common fractional unit, addition and subtraction of, 146 Franc. 213, 214 Freezing point, 116, ,363 Freezing temperatures, 116, 364. 365 Freight and passenger trains, 128-130 French notation. 18, and note Furnishings, house and, 78-81 Furrow of given width, number of acres covered by one mile of, .161 G. C. D., 141 Gain and loss, 240-242 Gale, 14 (]rallon, dry gallon and liquid, 216 Gallon, number of cubic inches in, 113, 216 Gallon of milk, number of pounds in one, 204 Gallons In barrel, number of, 34, 228 Gallons In one cubic foot, number of (liquid), 266 Gases, critical temperatures of, 368 Gate, figure of, 184 Geographic mile or knot, 227 Geography, 37, 38, 68-70, 245-248 Geometry, constructive. Bee also con- structions. 98-105, 176-182, 272- 281, 289-293, 307-320 German notation, 18, and note Gettysburg address, Lincoln's, 224 Gill {unit), 216 (Jlobe, figure* of, 304 Gold and silver coins, fineness of United States, 55, 226 GENERAL INDEX 391 Gold and water compared, weight of, 165 Gold, value of one ounce of, 55, 115 Gold, value of one pound of, 161 Government 28, etc., 342 (graduation of thermometers, 365, 366 Grain (unit of weight) , 214 Grain rent, 10 Gram, 231, 232 Granite, weight of, 56 Gravity, speclHc, 207, 208 Great gross, 217 Great Pyramid, dimensions of, 316 Greatest common divisor, 141 Greatest common divisor by prime fac- tors, 143, 144 Gregorian calendar, 217 Grocer's price list, 4 Grocery clerk, problems of, 92 Gross. 217 Growth, effect of factory life upon, 197 Growth of people In helpht and in weight, 13, 121, 122, 196, 197 Growth of trees, 147 Growth of trees, lateral, 147 Growth of United States, territorial, 246 Guinea, 213 Hawk, speed of, 75, 206 Hay and of coal in ton, number of cubic feet of, 205 Height, See also altitude and elevation Height and weight, growth of people, 13, 121. 122, 196. 197 ., Height to lung capacity, relation of, 12, 206 Heights of boys and girls working and not working in factories, 197 Heights of mountains, elevations and, 37-69, 243, 247 Heights of persons, 13, 121, 122, 196, 197, 206 Heights of railroad cars, engines and tenders, 15 Hekto-, 230 Heller, 214 Hexagon (and figure), regular, 104,282, 326 Hexagon, sum of angles of. 292 High land, ratio of low land to, 246 Higher terms, reduction of fraction to, 140, 141 Horse-power, 26, 206 Horse, speed of, 206 Hour, 217 Hour circle of sun, 304 House and furnishings, 78-81 House and roof, figure of. 59 House, figure of end of, 183 House plans, 78-81 Houses, walls of, 79 Hundred-weight, 214 Hundredth. 191 Hundredths , measuring by. See also percentage and ratio. 123-125 Hurricane, 14 Hypothenuse, 319 lUlnois admitted. 223 Immigration into United States, 1900, 1901, 27, 64, 65 Imports of coffee into United States, 27, 56. 203 Imports of molasses into United States, 27 Imports of tea into United States, 27, 56, 203 Improper fraction, 151 Inch precipitation, 57, 193 Inch ; also square and cubic inch, 214, 215 Independence Hall, Philadelphia, figure of, 274 Indorsement, 261 Industrial products, 1897, 1902, values of, 40 Inequality, expression of, 349 Inscribed square, 182 Insurance, 336, 337 Insurance policy, 336 Integer, 151 Interest, 125, 126, 252-263 Interest, compound, 343-345 Interest, percentage and, 123-126, 235- 263 Interest, simple, 125, 126 Interest table. 257 Intersection, 99 Introduction, general, 1-16 Introduction to fractions, 135-140 Introduction to ratio and proportion, 135-137 Invert, 170 Isosceles triangle (and figure), 102, 103, 282, 284 Isosceles triangle, base of, 102 Joint efPect of forces, 173-175 .Julian calendar, 217 Jupiter, diameter of, 317 Kilo-, 230 Kite, figure of, 184 Knot or geographic mile (nautical unit of distance), 227 L. C. D., 147 Lakes, areas of, 248 Lakes, depths of, 248 Lakes, elevations of, 35, 37, 248 Land (and figure), section of, 119, 215, 311 Land area of the earth, 246 Land areas of divisions of United States, 201 Land, description of, 310-312 Land, figure of divided section of, 119, 311 Land, figure of tract of, 7, 24, 119, 133, 185, 306, 310, 311, 331-335 Land, measuring, 119, 120, 133. 134, 215, 231, 310. 312, 329-336 Land surveying, 310-312, 329-336 Lateral growth of trees, 147 Latitude, 299 Latitudes of seaports, 299 Leaders of locomotive, 264 Leap year, 217 Least common denominator, 147 ' Least common multiple. 149-151 392 RATIONAL GRAMMAR SCHOOL ARITHMETIC Leayes of trees, 195, 243 Legal time. 305 Length, measuring, 106-108, 214, 215, 230 Letters to represent numbers, use of, 345-369 Light to travel from moon to earth, time for, 55 Light to travel from sun to earth, time for, 55 Light, velocity of, 19, 51, 55, 206 Lighted {room), well, 81, 108 Lincoln's Gettysburg address, 224 Line, 274 Line, base, 310 Line, date, 305 Line, standard base, 310 Linear measure, 214, 215, 230 Lines, concurrent, 279-281 Lines, division of, 176 Lines, figure of parallel, 99, 100, 177, 275, 276, 288 Lines, parallel, 180 Link (unit) ; also square link, 215 Liquid and dry gallon, 216 Liquid measure, 216 Lira, 213, 214 Liter, 231 Literal numbers, 43 Literal numbers, subtraction of, 42, 43 Live stock, a week's receipts and ship- ments of. See also cattle, 36 Living, cost of, 3-6, 93-96 Living, expenses of, 3-6, 93-96 Load. Sec tractive force Local (sun) time, 300, 305 Locating places on the earth, 297-299 Locomotive (and figure). See also en- gines and tenders. 264-271 Locomotive, drivers of, 56, 264, 270 Locomotive, leaders of, 264 Locomotive, trailers of, 264 Locomotives, lengths and weights of, 129, 130. 264 Long division, 65-70 Long ton, 214 Longitude, 299, 300 Longitude and time. 300.305 Longitude and time, equivalents of, 305 Longitude at the equator, length of one degree of, 164 Longitudes of observatories, 301 Longitudes of seaports. 299 liongltudes, table of. 302 Loss, gain and, 240-242 Lot (of land) (and figure). 311 Lots (and figure), town block and, 76 Low land to high land, ratio of. 246 Lowest terms of fraction, 141, 142 Lumber measuring, 8 Lumber sold by board foot, 8 Lung capacities, heights and weights of men, 206, 207 Lung capacity, 12 Lung capacity to height, relation of, 12 Maker (of a note), 259 Manufactured products, 1897, 1902, values of, 40 Manufactures, Chicago, 70 Manufactures, New York City, 1800, 37 Map of time belts of United States, 303 Maps of United States, 29, 303 Map of the world, 296 Mark, 213, 214 Marking goods, 251 Mars, diameter of, 317 Maturity, date of, 259 Mean, 12, footnote Mean solar day, 118, 217 Mean solar year, 119, 217 Mean temperatures. 244, 364 Means, 172, 186, 187 Measure, 136 Measure, chest, 12 Measure, dry, 216 Measure, linear, 214, 215, 230 Measure, liquid, 216 Measure, numerical, 136 Measure, surveyors', 215 Measure, to, 105 Measurement, 105-126, 136, 138 Measurements, 31. 32 Measurements, physical, 11-14. 206, 207 Measures, metric, 229-232 Measuring. See also mensuration Measuring angles. 285-289 Measuring arcs. 285-289 Measuring brick work, 78-80, 308, 309 Measuring bulk. See measuring volume Measuring by counting. 217 Measuring by hundredths. See also percentage and ratio, 123-125 Measuring by money. See measuring value Measuring capacity, 112. 113, 216. 231 Measuring distance. 106-108. 214, 215 Measuring fire-wood, 215, 232 Measuring land. 119, 120, 133. 134. 215, 231, 310, 311, 329-336 Measuring length, 106-108, 214, 215. 230 Measuring lumber, 8 Measuring paper, 217 Measuring roofing, .3()7-.309 Measuring stone, 205. 215 Measuring surface. 108-112. 215. 231 Measuring temperature. 116. 117 Measuring timp. 118. 119. 216 Measuring value. 105. 106. 213, 214 Measuring volume. 112, 113, 215, 231, 277, 278, 312317 Measuring weight. 114-116. 214. 231.232 Measuring wood (not fire-wood), 8 Median, 176 Melting temperatures, 364, 367 Members of equation, first and second. 97 Mensuration. See also measuring, etc., 306-317 Mensuration of circle, surface, 210, 211 Mensuration of circumference, 204, 205 Mensuration of cone (right circular), volume. 314 Mensuration of cylinder (right circular), lateral surface, 313 Mensuration of cylinder (right circular), volume. 313 Mensuration of oblique parallelepip.ed, volume, 277, 278, 812 GENERAL INDEX 393 Mensuration of oblique prism, volume, 277, 278 Mensuration of pail, capacity, 11 Mensuration of parallelepiped (and fig- ure), (rectangular), volume, 58, 112, 113, 312 Mensuration of parallelepiped (oblique), volume. 277, 278, 312 Mensuration of parallelogram, surface, 110. Ill, 179 Mensuration of prism (triangular) (and figure), volume, 277, 316 Mensuration of prism (oblique), volume, 277, 278 Mensuration of prism (right), volume, 277, 278 Mensuration of prism (square), volume, 112. 113, 277. 278, 312 Mensuration of pyramid (and figure) (triangular), volume, 316 Mensuration of ratio of circumference to diameter. 204, 205 Mensuration of rectangle, surface, 31, 32, 58, 108-110, 133, 134, 179 Mensuration of rectangular parallele- piped, volume, 58, 112, 113, 312 Mensuration of right circular cone, vol- ume, 314 Mensuration of right circular cylinder, lateral surface, 313 Mensuration of right circular cylinder, volume. 313 if ensu ration of right prism, volume. 277. 278 Vlensuration of sphere, surface, 315 ilensuration of sphere, volume, 317 IdLensuratlon of square-cornered box, vol- ume, 57, 58, 112, 113. ilensuration of square prism, volume, 112, 113, 277, 278, 312 Mensuration of square, surface, 108, 179 Alensuration of trapezoid, surface, 306, 307 Mensuration of triangle, surface. 111, 179 Mensuration of triangular prism (and figure), volume, 277, 316 Mensuration of triangular pyramid (and figure), volume, 316 Mercury, diameter of, 317 Meridian, 297, 300, 304, 310 Meridian, prime, 297, 300, 304 Meridian, principal, 310 Meteorology. See also weather, 242-244 Meter ; also square and cubic meter, 229-231 Methods of shortening and checking cal- culations, 369-374 Metre. See Meter Metric measures, 229-232 Metric system and equivalents In United States system, 229-232 Metric system, history of, 229, 230 Metric ton, 232, 234 Mile or knot, geographic. 227 Mile ; also square mile, 214, 215 Mile, statute. 214. 227 Mileage and numbers of employees of street railroads,' 65 Mileage of United States, 1900, rail- road, 816 Milk, number of pounds In one gallon of, 204 Milkings, weights of, 16, 75, 204, 241 Mill, 213 Milli-, 230 Million. 18, and foot-note Minnesota quarries, 228 Minus, 33 Minuend, 33 Minute of angle. 287, 289 Minute of arc, 287, 289 Minute of time, 217 Mixed decimal, 193 Mixed number, 151 Model and development of cone (right circular), 314 Model and development of cube (three- inch), 274 Model and development of cylinder (right circular), 313 Model and development of flat prism, 276 Model and development of oblique paral- lelogram prism, 277 Model and development of prism (flat), 276 Model and development of prism (ob- lique parallelogram), 277 Model nnd development of prism (right), 276 Model and development of prism (square), 275 Model and development of prism (tri- angular), 277 Model and development of pyramid (tri- angular), 316 Model and development of right circular cone, 314 Model and development of right circular cylinder, 313 Model and development of right prism, 276 Model and development of roof. 307, 309 Model and development of room, 109 Model and development of square prism, 275 Model and development of three-inch cube. 274 Model and development of tower, 309 Model and development of triangular , prism, 277 Model and development of triangular pyramid, 316 Molasses into the United States, Imports of, 27 Money, 105 Money, kinds of, 213, 214 Month, calendar, 217 Moon, diameter of, 317 Moon, mean distance from earth, 19, 206. 331 Moon, velocity of, 83 Mountain time, 303 Mountains, elevations and heights of, 37, 69, 243. 247 Movement of wind, 242, 243 Multiple, 148, 149 Multiple, common, 148 Multiple, least common, 149-151 Multiplicand, 44 Multiplication by factors, 61, 52 394 RATIONAL GRAMMAR SCHOOL ARITHMETIC Multiplication by fractional numbers. 54 Multiplication by number near 10, 100, 1000, etc., 52 Multiplication by 10, 100. 1000, etc., 52 Multiplication by 25, 50, 12 V^, 75. 500, 250, pp. 52, 53 Multiplication, checking, 53, 54, 369, 371-373 Multiplication compared, dirision and, 61-63 Multiplication (of common numbers), 43-60 Multiplication of decimals. Sec decimals Multiplication of fractions. Bee frac- tions Multiplication, short methods for, 51-53, 192, 371-373 Multiplication table, 45. 46 Multiplication when some dibits of mul- tiplier are factors of others, 53 Multiplier, 44 Myria-, 230 Name value of diffit, 17 National League, one season's record of, 238 Nature study, 19.*). 106 Negative, 116, 117, 175 Neptune, diameter of, 317 New England. 29 N-gon, sum of angles of, 203 Nines, casting out the, 28, 53, 54, 75 Non-terminating decimal, 209 Noon, 217 North America, area of, 247 Notation and numeration, 17-21 Notation, Arabic, 20 Notation, change of, 21 Notation, decimal, 19, 190, 191 Notation, English, 18, and note Notation, French, 18, and note Notation, German, 18, and note Notation of decimals, 190, 191 Notation. Roman, 20, 21 Notation, United States, 18, and note Note, promissory. 259, 260 Notes, coupon, 345 Notes, discounting, 260, 261 Number, composite, 52. 143 Number, denominate, 213 Number, even, 143 Number, mixed, 151 Number, odd, 143 Number of decimal places, 198 Number, prime. 52, 143 Number, square of, 321 Numbers, common uses of, 127-134 Numbers, compound denominate, 213- 225 Numbers, divisibility of, 73, 74 Numbers for individual work, 41, 42 Numbers, literal, 43 Numbers, periods of, 18 Numbers, reading. 19, 20 Numbers, subtraction of literal, 42, 43 Numbers, use of letters to represent, . 345-369 Numliers, writing, 20 Numeral, Arabic, 20, 21 Numeral, Roman, 20, 21, 118, 119 Numeration, notation and, 17-21 Numeration of decimals, 191, 192 Numerator, 139 Numerical measure, 136 Oblique parallelepiped, volume of, 277, 278, 312 Oblique prism (and figure), volume of, 277, 278 Observatories, longitudes of, 301 Obtuse angle (and figure), 289 Ocean depth, greatest, 69 Octagon, sum of angles of, 293 Octant, 286 Odd number, 143 One-brick wall, 78, foot-note Operations, order of, 159 Opposite or vertical angles, 288 Order of factors. 160 Order of operations, 159 Ounce, 214 Ounce of gold, value of, 55, 115 Pacific time. 303 I*ail, capacity of. 11 Taper-folding (creasing), 278-281. See also constructions Paper, measuring, 217 Par, above, 341 Par, at, 341 Par, below. 341 Parallel, 298 Parallel lines, 180 Parallel lines and line cutting them, angles formed by, 288 Parallel lines, figure of, 99, 100, 177, 275 276 288 Parallel* ruler (and figure), 177, 178 Parallelepiped, volume of oblique, 277, 278, 312 Parallelepiped, volume of rectangular, 58. 112, 113, 312 Parallelogram, 110, 283 Altitude of. 111 Area of, 110, 111, 179 Base of, 111 Compared (and figures), circle and. 210 Figure of, 110, 111, 179, 292, 203 Properties of, 293 Sum of angles of, 292 Partial payments, 261-263 Partial l»ayments, United States Rule for, 262 Paving, 77, 78 I»ayee, 259 Payments, partial, 261-263 Peck, 71, 216 Pencil-foot of compasses, 98 Pen-foot of compasses, 98 Penny, 213 Pennyweight, 214 Pentagon (and figure), 282, 826 Per annum. 252 Per cent, 124, 235, 236 Percentage, 123-125, 235-251 Percentage and interest, 123-126, 235- 263 Percentage, applications of, 336-345 GENERAL INDEX 395 Perch of stone, 205, 215 Perlgon, 287, 289 Perimeter, 105, 154, 282-284 Periods of numbers, 18 Perpendicular, 278 Perpendicular bisector, (and figure), 101, 176 Personal property, 338 Persons, heights and weights of, 13, 121, 122. 106, 197, 206 Pfennig, 214, 277 Physical measurements, 11-14, 206, 207 Pi, value of, 205, 370 Pigeon, speed of carrier, 206 Pin-foot of compasses, 98 Pint, 216 Piston, figure of, 211 Place value of digit, 17 Places, number of decimal. 198 Planets, diameters of, 317 Play-house, figure of, 183 Plotted curves, 86, 120, 122, 267, 368 Plotting observations and measurements, 120-123 Plus, 22 Point (decimal). 190 Point (geometric), 274 Pointing off product of decimals, 198, 199 Policy, face of, 337 Policy, Insurance, 337 Population, density of, 70 Population of Switzerland, area and, 69 Population of United States, 1790-1900, 123 Population of United States, 1900, dis- tribution of. 29-31 Populations of cities, 28, 41 Populations of countries, areas and. 70 Populations of states, territories, etc., 1890, p. 39 Populations of states, territories, etc., 1900, p. 30 Positive, 175 Positive and negative readings of ther- mometer, 116, 117, 365, 367 Pound, 214 Pound, comparison of avoirdupois and troy, 116 Pound sterling, 213 Pounds in one gallon of milk, number of. 204 Power (of a number), 192 Practical applications of proportion, 188-190, 327-336 Precloitatlon in various places, 25, 56- 59, 193, 194, 244. See also weather Precipitation, inch, 57, 193 Preferred stock, 342 Premium, 337 Present worth. 260 Pressure of air. 127. 128 Pressure of wind, 14, 15 Price. Bee marking goods Prices of provisions. 4 Prime factor. 142. 143 Prime meridian, 297, 300, 304 Prime number, 52. 143 Prime to each other {phrase) ^ 142 Principal, 248. 252 Principal meriaiftn, 310 Principles for using the equation. I 350, II 350, III 351, IV 352, V 352, VI 366 Principles of decimals, I 193, II 199, III 203 Principles of fractions, I 141, II 144, III 146, IV 150, V 152, VI 152, VII 153, VIII 158, IX 166. X 170, XI 170 Prism (and figure), volume of triangu- lar, 277, 316 Prism, figure of oblique (oblique pile), 278 Prism, figure of right (square pile), 277 Prism, model and development of fiat, 276 Prism, model and development of oblique parallelogram, 277 Prism, model and development of right, 276 Prism, model and development of square, 275 Prism, model and development of trian- gular 277 Prism, volume of oblique, 277, 278 Prism, volume of right, 277, 278 Prism, volume of square, 112, 113, 277, 278, 312 Prisms, comparison of, 276-278 Problem, statement of, 44 Problems of grocery clerk. 92 Product, 44 Products, table of, 45, 46 Products of means and of extremes, 187 Products of sums and differences of lines, 293-295 Products, 1897, 1902, values of manu- factured and industrial, 40 Promissory notes, 259, 260 Proper fraction, 151 Property, personal, 338 Property, real. 338 Proportion. 136, 137, 186-190. Bee also similar triangles Proportion, practical applications of, 188-100, 327-336 Proportion, terms of, 186 Protractor (and figure), 286-289 Provisions, prlcrs of, 4 Pull. Bee tractive force Pulse, rapidity of. 51 Pure decimal. 193 Pyramid (and figure), volume of trian- gular, 316 Pyramid, dimensions of Great. 816 Pyramid, model and development of tri- angular, 316 Q., 232 Quadrant. 229. 285. 289 Quadrilateral (and figure), 283, 306, 307 Quadrilateral sum of angles of, 292 Quadrillion, 18. and foot-note Quarries. Minnesota, 228 Quart, 216 Quintal. 232 Quintilllon, 18, and foot-note Quire, 217 Quotations and transactions, stock and bond, 341 396 RATIONAL GRAMMAR SCHOOL ARITHMETIC Quotient, 62 Radical sign, 322, 327 Radius, 98, 107 Radius of earth, mean, 19, 316 Radius of sun, mean, 19 Rail, weight of one yard of steel, 54, 64, 67 Railroad cars, engines and tenders, heights, lengths, and weights of, 15, 64, 129, 130, 266 Railroad cars, tractive force for street, 67, 68 Railroad data, street, 65 Railroad mileage of United States, 1900, 316 Railroad trains, problems on, 128-130 Railroad trains, tractive force for, 129, 130, 266-271 Railroads, mileage and number of em- ployees of street, 65 Rain and snowfall. See also precipita- tion, 193, 194 Rain gauge, figure of, 56 Rainfall. See also precipitation and weather, 56-59 Rainfall, Inch, 57, 193 Rains, numbers of, 25. See also precipi- tation Range (in land surveys), 310 Range of temperature, 86 Rate, 236, 252 Rate of running, average, 132 Rate per cent, 236, 252 Ratio, 135, 136. See also similar tri- angles Ratio and as equal parts, fraction as, 138-140 Ratio of circumference to diameter, 204, 205 Reading numbers, 19, 20 Reading of thermometer, 363 Readings of thermometer, positive and negative, 365, 367 Real property, 338 Ream, 217 Reaumur thermometer, 363-369 Reciprocal, 170 Rectangle (and figure), 110, 179, 282- 284 Rectangle, area of. 31, 32, 108, 179 Rectangle, properties of, 293 Rectangular parallelepiped (and figure), volume of, 58, 112. 113. 312 Reduction, see also equivalents Reduction, ascending, 225 Reduction, descending, 225 Reduction of common fraction to deci- mal, 208, 209 Reduction of decimal to common frac- tion, 192. 193 Reduction of fractions, 140-142. 146, 147, 151-153, 192. 193. 208. 209 Reduction of fractions to higher, lower, and lowest terms, 140-142 Relation signs, 350 Remainder, 33 Rent, cash, 10 Rent, grain, 10 Repeteod, 209 Rhombus (and figure), 282, 283 Rifle ball, velocity of, 206 Right angle (and figure). 278, 289 Right circular cone, volume of, 314 Right circular cylinder, lateral surface of, 313 Right circular cylinder, model and de- velopment of, 313 Right circular cylinder, volume of, 313 Right prism, model and development of, 276 Right prism (square pile), figure of, 277 Right prism, volume of, 277, 278 Right section, 211 Right triangle (and figure), 179, 319, 320 Right triangle, relation of sides of, 320, 321, 325 River basins, areas of, 37, 247 Rivers, lengths of, 247 Rivers, rapidity of, 206 Road wagon, tractive force for, 67, 199- 202, 269, 270, 348 Rod (unit) ; also square rod, 214, 215 Roman notation, 20, 21 Roman numeral. 20, 21, 118, 119 Roof, figure of house and, 59 Roof model and development of, 307 Roofing and brick work, 307-309 Roofing, square of, 307 Room, model and development of, 109 Root, short method for square, 374 Root, square, 322-326 Roots, cubes and cube, 326, 327 Roots, squares and square, 321-326 Ropes, data concerning, 360 Ruble, 213, 214 Ruler (and figure), parallel, 177, 178 Running time, 131 Rye, area and yield of Kentucky, 70 Saturn, diameter of, 317 Scale drawings, 1, 2, 7, 24, 76, 79 81, 108, 119, 133, 134, 183-195, See also models and plotting curves Scale drawings of familiar objects, 183- 186 Scalene triangle (and figure), 103, 282 Scales, figure of, 97 Schedule, train, 131 School data, Cuban, 26 School data for largest cities of United States, 41 School data for states, territories, etc., 30, 37 School grounds, figure of, 185 School house and grounds, figure of, 185, 186 School room, figure of, 1 Score. 217 Scruple, 214 Second of angle, 287, 289 Second of arc, 287. 289 Second of time, 217 Section of land (and figure), 119, 215, 311 Section of land, figure of divided, 119, 311 Section, right, 211 Sextant, 286 GENERAL INDEX 397 Shape, likeness of, 137 Share of stock, 341 Shilling, 213 Shingling, 307. 308 Ships. See battleships Short division. 63. 64 Short methods for addition, 360, 370 Short methods for division, 70-74, 101, 373 Short methods for multiplication, 51- 53, 192, 371-373 Short methods for square root, 374 Short methods for subtraction. 360-371 Shortening and checkine: calculations, methods for, 369-374 Sides of angle, 288 Sides of equation, left and right, 97 Sides of similar triangles, corresponding, 328 Signs, relation, 350 Signs: ' foot, feet; minute of angle and of arc, 79. 289 " inch, inches ; second of angle and of arc, 70, 289 -f plus, 22 = equal, equals, 22, 97 — minus, 33 X times, multiplied b.v, 44 -=- / — divided by, 62. 63 % hundredth, hundredths. 124. 236 <, > is less than : is greater than, 349 ± perpendicular to, 319 V square root of. 322 f cube root of, 327 Silver coins, fineness of United States gold and, 55, 226 Silver, value of one pound of, 55 Similar triangles. See also ratio and proportion, 137. 327-329 Similar triangles, uses of, 327-336 Simple decimal, 193 Simple Interest, 125, 126 Six per cent method, 125. 126, 252 Sleeping car, length of, 64 Soil, 195 Solar day, mean, 118, 217 Solar year, mean, 217 Solidus, 63 Sound, velocity of. 55, 83. 206 Sparrow, speed of. 206 Specific gravity, 207, 208 Sphere, area of surface of, 315 Sphere, volume of, 317 Spirometer (and figure). 11, 12 Square (and figure), 282, 284 Square, area of, 108, 179 Square, center of, 320 Square cornered box, volume of, 57, 58, 112 Square, inscribed, 182 Square of number, 321 Square of roofing. 307 Square prism, model and development of, 275 Square prism, volume of, 112, 113, 277, 278, 312 Square root, 322 Square root bv subtraction, 874 Square root» cnecking, 874 Square root, short method for, 374 Square roots, squares and, 321-326 Square unit, 31, 108 Squares and square roots, 321-326 Squares of units, tens, and hundreds, table of, 322 Standard base line, 310 Standard time, 303-305 Statement of the problem, 44 Statements in words and in symbols, 354-360 States, territories, etc., areas of, 30. 35 States, territories, etc., dates of admis- sion and populations (1890) of. 39 States, territories, etc., populations (1890, 1900), of, 30, 39 States, territories, etc., school data for, 30 Stature of persons. See aUo heights, etc., 196, 197 Statute mile, 214, 227 Steamboat, speed of, 206 Steel rail, weight of one yard of, 54, 64, 67 Steel, weight of one cubic inch of, 113 Stere, 232 Stock and bond quotations and transac- tions, 341 Stock certificate, 341 Stock, common. 342 Stock, preferred. .S42 Stock, share of, 341 Stocks and bonds, 341-343 Stone, measuring, 205, 215 Straight angle (and figure), 285 Street railroad cars, tractive force for, 67, 68 Street railroad data, 65 Street railroad mileage and numbers of employees, 65 Streets and blocks, figure of city, 306 Substituting, 267 Subtraction, checking. 369, 371 Subtraction compared, division and, 60, 61 Subtraction of angles, 290 Subtraction (of common numbers), 32- 43 Subtraction of decimals. See also deci- mals, 195-197 Subtraction of fractions. See fractions Subtraction of fractions with common fractional unit, 146 Subtraction of literal numbers, 42. 43 Subtraction, short methods for, 369, 371 Subtrahend. 33 Sugar production, 37 Sum, 22' Sum and difference of angles. 289-293 Sum of acute angles of right triangle, 292 Sum of angles of hexagon. 292 Sum of angles of n-gon. 293 Sum of angles of octagon, 293 Sum of angles of parallelogram. 292 Sum of angles of quadrilateral, 292 Sum of angles of trianele. 291 Sums and differences of lines, products of, 293-295 Sun, diameter of, 317 398 RATIONAL GRAMMAR SCHOOL ARITHMETIC Sun, mean distance from earth to, 19, 206, 332 Sun, mean radius of, 19 S^un, rising and setting of, 223 Sun time, local, 300, 305 Supplemental angles, 291 Surface measuring, 15, 31, 32, 108- 112, 132-134, 215, 231, 306-312 Surveying land, 310-312. 329-336 Surveyors' measure, 215 Switzerland, population and area of, 69 Symbol. See sign Symbols, statements in words and in, 354-360 Tax, 338 Taxes. See also assessment, 338, 339 Tea into United States, imports of, 27, 56, 203 Telegraph wire, cost and weight of, 45, 55 Temperature. See also thermometer aft<I weather, 85-87, 244 Temperature, measuring, 116, 117 Temperature of a man's body, normal, 368 Temperature, range of, 86 Temperatures, boiling, 116, 364, 368 Temperatures, critical, 368 Temperatures, freezing, 116, 364 Temperatures, mean. 244, 364 Temperatures, melting, 364, 367 Temperatures of gases, critical, 368 Ten-thousandth, 191 Tenant farmer, 10 Tenders, heights, lengths, and weights of railroad cars, engines, and, 15, 64, 129, 130. 264 Tenders, water and coal capacity of locomotive, 264 Tenth (decimal), 191 Terms of fraction. See also lowest terms, 139 Terms of proportion, 186 Territorial growth of United States, 246 Territory of United States, area of ac- quired, 246 Tests of divisibility of numbers, 73, 74 Thermometer (and figure), centigrade. See also weather, 363-369 Thermometer (and figure), Fahrenheit, 85-87, 116, 363-369 Thermometer (and figure), Reamur, 363-369 Thermometer, graphically, laws of, 368, 369 Thermometer, positive and negative readings of, 365. 367 Thermometer, ransre of, 86 Thermometer, reading of, 363 Thermometer. See also weather. 85-87, 116, 363-369 Thermometers, equivalent readings of, 367-369 Thermometers, graduation of, 365, 366 Thousand. 18 Thousandth, 191 Time belts of United States, map of, 303 Time, central, 303 Time, eastern, 303 Time, equivalents of longitude and, 305 Time, legal, 305 Time, local (sun). 300, 305 Time, longitude and, 300-305 Time, measuring, 118, 119, 216 Time, minute of, 217 Time, mountain, 303 Time, Pacific, 303 Time, second of, 217 Time, standard, 303-305 Time, sun, 300, 305 Time to run (phrase), 259 To the weather (phrase), 307 Ton, 214 Ton, long, 214 Ton, metric, 232 Ton, number of cubic feet of hay and of coal in, 205 Tonneau. 234 Tower, model and development of, 309 Town block and lots (and figure), 76 Township (and figure), 120, 215, 310, 311 Tract of land, figure of, 7. 24, 119, 133, 185, 306, 310, 311, 331-335 Tractive force for railroad trains, 129, 130, 266-268, 270, 271 Tractive force for road wagon, 67, 199- 202, 269, 270, 348 Tractive force for street railroad cars, 67, 68 Trade discount, 249, 250. 339, 340 Trade, 1891, 1901, United States export, 40 Trailers of locomotive, 264 Train despatcher's report, 130-132 Train schedule. 131 Trains at Clhicago, number of daily, 63 Trains, freight and passenger, 128-130 Trains, problems on railroad, 128-130 Trains, speed of express. 82, 83 Trains, tractive force for railroad, 129, 130, 266-271 Transportation problems, applications to. See also tractive, force. 264- 271 Trapezium (and figure), 282 Trapezoid (and figure), 184, 282, 283, 306 Altitude of, 306 Area of. 306, 307 Bases of, 306 Dimensions of, 307 Trees, lateral growth of, 147 Trees, lateral growth of, 147 Trees, leaves of. 195, 243 Triangle, altitude of. Ill Triangle (and figure), equilateral, 102, 103, 282, 284, 326 Triangle (and figure), isosceles, 102, 103, 282.. 284 Triangle (and figure), right, 179 Triangle (and figure), scalene. 103, 282 Triangle, area of, 110, 111, 133, 134, 179 Triangle, base of isosceles, 102 Trlansrle, relation of sides of right, 320, 321. 325 Triangle, sum of angles of. 291 Triangles (and figures) and their -uses, similar, 137, 327-336 GENERAL INDEX 399 Trlanfi:Ies (and figures), uses of the 30** and 60% and of the 46"', right, 180- 182 Triangular prism (and figure), volume of, 277, 316 Triangular prism, model and develop- ment of, 277 Triangular pyramid (and figure), vol- ume of, 316 Triangular pyramid, model and develop- ment of, 316 Trillion, 18, and footnote Trisect, 178 Troy weight, 214 True discount. 260 Twilight, 224 Unit, 18 Unit, concrete, 213 Unit, fractional. 107« 138 Unit, square, 31, 108 United States, land areas of divisions of, 201 United States, maps of, 29, 303 United States notation, 18, and note United States Rule for Partial Pay- ments, 262 United States, territorial growth of, 246 United States, 1790-1900, population of, 123 United States, 1900, distribution of pop- ulation of, 29-31 Uranus, diameter of, 317 Use of letters to represent numbers, 345-369 Uses of numbers, common, 127-134 Uses of similar triangles, 137. 327-336 Uses of the equation, 848, 363-369 Uses of the 30** and 60°, and the 45% right triangles, 180-182 Valuation, assessed, 338 Value, face, 259 Value, measuring, 105, 106, 213, 214 Value of digit, name, 17 Value of digit, place, 17 Value of X, 44 Values of cattle, weights and, 06 Venus, diameter of, 317 Vertex, 176, 274, 288 Vertex of angle, 176, 288 Vertical angles, opposite or, 288 Vertices. Plural of vertex Volume, 277, 312 Volume, measuring. 112. 113, 215, 231, 277, 278, 312-317 Volumes, 312-317 Wagon, tractive force for road, 67, 199- 202, 269. 270, 348 Walls of house, 79 Watch, figure of, 118 W^ater area of earth, 246 Water at greatest density, temperature of, 231 W^ater. compared, weight of gold and, 165 Water, weight of one cubic foot of, 165, 266 Water, weight of one cubic Inch of, 164 Weather. See barometer, heat, light, meteorology, precipitation, rainfall, temperature, thermometer, wind' Weather Bureau stations, elevations of, 243 Weather, to the (phrase), 307 Week, 217 Weight, apothecaries*, 214 Weight, avoirdupois, 214 Weight, growth of people, height and, 13, 121, 122, 196, 197, 206 Weight, measuring, 114-116, 214, 231, 232 Weight of one cubic toctt of varlouB things, 56. 64, 82, 114, 161. 165, 207. 208, 235, 266, 313 Weight of one cubic inch of steel, 113 Weight to lung capacity, relation of, 17 Weight, troy, 214 Well lighted (room), 81. 108 Wheat yield and area of Dakotas, 70 Wild duck, speed of, 206 Winchester bushel, 216 Wind movement. Bee also weather, 242, 243 Wind pressure, 14, 15 WMnd, velocity of, 14, 15, 242 Winds, names of, 14 Wire, cost and weight of fence, 7, 8, 133 Wire, cost and weight of telegraph, 45, 55 Wood (not fire-wood), measuring, 8 World, map of, 296 Worth, present, 260 Writing numbers, 20 Yard: also square and cubic yard, 214, 215 Yards In one mile, number, 54, note Year, 217 Year, calendar, 217 Year, common, 217 Year, leap, 217 Year, mean solar, 119, 217 Zero, 17 ANSWERS TO THE RATIONAL GRAMMAR SCHOOL ARITHMETIC Page 6 8. $.07 4. $.18 6. $2.75chanise 9. $904.56 10. ]\)th part Page 6 11. $62.40 12. 1897— $79.86 1898— $80.40 1899— $95.28 1900- $95. 67 1901— $101.59 13. $21 . 73 14. No. Breadstuffs, $6.49 Meat $2.33 Dairy and Gar- den $2.88 Other Food .64 ClothinK .90 Metals 3.81 Misc. 4 . 68 IT.Bread $15.01 Meat $8 . 04 Dairy $13.67 Other Food $9.01 Clothing $15. 57 Metals $14.54 Mi»c. $14.73 19. 1897— 98— $.54 1898— 99- $14.88 1899— 1900— .39 1900— 01— $5.92 Page 7 1. (a)160rods (b) i mile (0) 640 rods of wire 3. 320 rods wire Page 8 6. 640 posts 7. $160 8. $49.92 9. $71.25 Page 8 10. $122.07 18. 55 13. 550 19. $50.37 16. 5: 10; 12 17.20; 6; 7^; 16; 8; 6f. 18. 4; 8 Page 9 19. 6f: lOf 20. 13i: 16; 24 21. 6; 30 22. $5.94 23. $109.77 24 $181.29 25. $529.66 Page 10 1. $2,105.60; $13.16 2. $8.16; $1,305.60 3. $800 4. for ^ of ail crops $192 more • profitable 6. Corn more pro- fitable by $384 6. Wheat $512 more Corn $896 more 7. Grain $160 more 8. Cash rent $96 more 9. 3264 bu. 10. $1103.64 11. $729.64; $374 Page 11 1. 121.70; 304.- 25; 486.80 cu. in. 2. 91.275; 106- 487; 129.306; 140.71 3. 60.85; 7.606; 64.653; 76.06; 79.86; 83.668 4. 121.70 cu. in. Page 14 1. 35; 38901; 11671* 2. 662,500 lbs. 8. 46681 lbs. Page 15 4. 21787* lbs. 5. 184078.125 lbs 6. 154461; 1634,')i 7. 49021.875 lbs. 8. 1683 lbs. Page 16 1.44.5; 47.8; 92.3 2.N0V. 7.P.M. 2.9 Page 18 1.45.284; 6.934; 321,113; 6,005 2.9.010; 10.009; 600,6«0; 5.050.- 003; 80.080 Page 20 1.4.213,122.607; 45,450,327.001 2.718.200.000.- 000; 190,402; 3,- 000.124; 15.009; 90; 186,000.000.- 000; 147,100.100 Page 23 1. 242 2. $12.10 8. $1.85 4. 35 5. $.60 6. 233 mi. 7. 12 yd 8. 21 yd.s. 9. 19 yds. 10. 44 spools. 11. 4k doz. Page 24 12. 34 13. 19 lbs. 14. 73 lbs. 15. 265 lbs. 16. 20 bu. 17. 12 bu. Page 24 18. 35 doz. 19. 23 lbs. 2. 1567 Page 25 8. 1031 4. 66 5. 287 6. Chicago, 26 10.46 in. Des Moines, 29 11.33 in. Detroit, 32 0.62 in. Kansas City, 28 0.21 in. Omaha, 29 10.12 in. St. Louis, 23 11.29 in. Page 26 7. 2876 ft. 8. 3967 9. 88,304 10. 88,995 11.3667 teachers and schools 172.273 schol- ars 12. Immigrants, 1901—264,870 male Immigrants, 1901— 123.665 female Total. 388,535 Immigrants, 1900—243,091 male Immigrants 1900—117,058 female Total. 360,149 Page 27 13. Week No. 1st.. 1,798,798 2nd.. 1,823,561 3rd., 1.814,340 4th.. 1,820,085 5th., 605.388 Total, 7.862,172 400 ANSWERS 401 Page 27 14. 148,890.876 lbs 16. 245.083,133 16. $61,180,966 Page 28 17. 2,050.000,000 18. 9.477.398 Page 29 19. 17.436.267 20. 26.^1 3.665 21. 22,965,741 1. 5,592.217 2. 26.333,004 3. 4.001,349 4. 84,141.954 Page 30 6. 3.740.381 Page 31 6. 765.855 7.620,290 N. Cen- tral 8.Rhode Island 1.250 Texas 265.780 New York 7.268,894 Nevada 42.335 Nevada more than twice size of New Yock. 9.956.731 5.823- 019 10. 15,341,220 11. 28.686.008 18. Population 5.592.217; area, 66,465 sq.mi. Population 15.- 454,678 area 102.200 sq.ini. 14.956.731 ; 2,676,- 509 2. 50:5 Page 32 3. 6 4. 9 6. 6:15 6. a 7. 15z 8.15 9's: 15 12's: 15 x: 15z 9.1. 17 a, II. 14 a, III. 73 a. IV. 1462 z.V. 125 y, VI. 148 b, VII. 23 m. VIII. 241 n. IX. 197 c, X. 191 d. XI. 1981 X. XII. 1877 X. Page 33 1. 238 mi. Page 34 3. 408 6. 437 7. 7 yd. 8. i pk. Page 34 9. I yd. 1.25i gal: 23 gal: 18 gal: 15 gal: 12^ gal: 10 gal. 2. 7T)u. 8. 15.000 Page 36 4. 1215 6.(1) $7428.56 (2) $2675.12 (3) 299,189 yd. 6.(1) 5,037,273: (2) $1282.78 (3) 292i 7.Depo8its $373.- 80 8. 30,466 9. 3598 ft. 10.68,526 sq. mi. 11.47,259: 1955: 201.649: 71.322 12. 12,404: 254: 24.058: 14.998 18. 34.855: 1701: 177,591: 56.324 14.A11 less than previous week. Cattle, 15.929 less Calves, 1.859 less Hogs, 71.777 less Sheep. 33,206 less Corres. week 1900 Cattle, 6.706 less Calves. 133 more Hogs, 10,545 more Sheep, 16,276 more Corres. week. 1899 Cattle, 14,537 more Calves. 620 more Hogs. 41,925 more Sheep, 9,604 more 16. All less than previous week. Cattle, 6,295 less Calves, 70 less Hogs. 5.428 less Sheep. 9.431 less Compared to corresp. week 1900 Cattle, 6,034 less Calves, 98 less Hogs, 483 more Sheep, 666 more Compared to Corresp. week 1899 Cattle, 2,848 more Calves, 44 more Hogs. 7,421 more Sheep, 11,816 more 16.104,100 cattle 393.000 hogs 104.800 sheep 17.Chicago great- est in cattle: 18,- 000 more than Kansas City, Page 36 30,500 more than Omaha, 36.600 more than St. Louis. Chicago great- est in Hogs: 112,600 more than Kansas City. 136.400 more than Omaha. 163,500 more than St. Louis. Chicago great- est in sheep : 56,400 more than Kansas City. 58,500 more than Omaha. 65.500 more than St. Louis, Page 37 18.As compared with the pre- vious day . Cattle deer. 40,500 Hogs deer. 87,800 Sheep deer. 68,800 Corresp. week 1900 Cattle deer. 7.400 Hogs incr. 54,100 Sheep incr. 19,800 Corresp. week 1899 Cattle incr. 12,400 Hogs incr. 187,800 Sheep incr. 23,.500 Corresp. week 1898 Cattle deer. 24.300 Hogs incr. 28,400 Sheep incr. 9,200 Corresp. week 1897 Cattle deer. 38,900 Hogs incr. 19,800 Sheep deer. 800 1. 12,043 ft. 2. 13,258 ft. 8.1,250,000 sq.mi. 4.43.950 sq. mi. 6. Philippines smaller than Texas by 151.- 370 sq. mi. Philippines larger than Illi- nois by 57.760 sq. mi. Philippines larger than R. I by 113,160 sq.m 6. 5,150.235 7. 658,678T. 8. $120,706,122 9. 123 years 10.264,530 sq. mi. 11. 7,226,559 Page 38 12. 1,202,898 14.So. Atlantic by 113.870 sq. mi; No. Central by 145,565 sq. mi. Western by 567.245 sq. mi. 16.No. A 1 1 a n- t i c, 10.603,415 more; No. Cent. 12.253.047 more So. Central 9,988,608 more 16.563,748; 1,191.- 093; Maine ex- ceeds by 627,- 345 18.2,196,681; dif. in total pop. larger than thus by 250,663; dif. in total pop. be- ing 2,447,344 20.Mainc 33,580 N.H. 35,058 Vt. 11,219 Mass. 567.303 R. I. 83,050 Conn. 162,162 N. Y. 1,271.041 N. J. 438,736 Penn. 1,044,101 Total. 3,646,250 24.W. N. Atlan- tic incr. 1,861.- 506 over E. N. Atlantic.S. S. Atlantic incr. 376,696 over N. S. Atlantic. E. N. Central incr. 1,056.766 over W. N. Central. E. S. Central incr. 1679,088 over W. S. Cen- tral. W. Wes- tern gained 155,- 803 more than E. Western. W. Western gained 416,576 more than MiddleWestern. E. Western gained 260,873 more than Middle Western 1. $1,871,568,787 2.Un. Kingdom, + $116,471,003 Ger. +94,352,391 Can., +65,809,640 Neth. +54,382.038 Mex., +21,400,198 Italy, +19.599,197 British Australia, + 17,004,883 British Africa, + 21,483,098 402 RATIONAL GRAMMAR SCHOOL ARITHMETIC Pase 38 Jap., +17,323.093 Bra., -3.928,245 Arg. +9.207,733 Ru8. +1,104.510 Page 41 S.Ag. Implements + $1,832,143 Books, maps, etc.. + 517.837 Carriages and cars + 833,448 Cop. ing. + 166.855 Cot. clth. -1.114.683 Cot.mfs. +650.981 Cycles and parts of -241,087 Bld'r'a hardware + 357,616 Swg. machines, + 112,954 Oth. Machinery, -328,378 Corn, -302.141 Wheat +1,220.799 W. flour -1,777,158 Coal, -1.614,679 Cotton +1,882.526 Fr. & n. + 778.676 Furs and fur skins + 471,630 Cotton Seed Oil + $214,619 Beef, salted or pkl. + 32.783 Bacon, +192,408 Hams, +30,879 Page 42 l.Pop. 8,074,559 Pop. 5,974,091 Sch. Enroll., 1,- 213.519;no. tch. 27,123; school ex.,$40.118, 540 2.Pop., 1900, 2,582.578; Pop., 1890, 2.003.922; Sch. Enroll., 367.287; no. teachers, 8.261; sch. expend . $9,756,432 Pop., 1900, 1,912.334; pop. 1890, 1.496,982; sch. Enroll. 310,325; sch. ex., $6,534,163 no. teachers, 7,356 3. 120,415 4.242; 6,261; 267; 3,933 5.9.084; 26,196; 32,161 Page 42 6. $2,138,158; $- 730,333; $654,- 029; $525,965; $693,334 7. More than half; less than half. S.Pop.. 1900. 13.- 317.403; pop.. 1890, 10,029,- 377; sch. enroll. 2,019.428; no. teachers, 45.- 663; sch. ex., $60,176,405 Incr. pop. 3,- 288,026 1. 4 2. 7 8. 6 9s 4. 6 c's Page 48 5. 9x, 18y; 6z; 29z; 686a; 142a 6. 120x aq. ft. 7. 6x cts. 8.6; 35; 95; 26; 8; 5; 5; 6; 9; 12; 2; 100; 32; lOi; 72 Page 44 2. 126.720 8. $2.04 4. 768 Page 46 5. 258,300 6. $20,664 8. 512 9.27,664: 1400: 66x 10.135,597: 231y: 231 a Page 47 1. Receipts, $512.18; Ex- penses. $128.23; profit. $383.95 per 40 acres ; $9.^98 per acre Page 48 2. Receipts. $296; expenses. $106.80; profit per 20 acres, $188.20; profit per acre, $9.41 3. Receipts, $1803.20; exp.. $704.40; profit for 30 acres, $1098.80; pro- fit per acre, $36.62f Page 49 4. Receipts, $638.76; exp., $166.69; profit 60 acres, $472.06; profit per acre, 9.44/0 6. Receipts, $114; expen. 39.25; profit $74.76 for 10 acres; profit per acre, $7,476 6. Receipts, $368.00; ex., 122.26; profit $228.15 for 10 acre lot; profit $22,816 for 1 Page 60 7. Expendi- tures, $766.75 8. Receipts, $3713.63; exp. $2034.37; pro- fit, $1679.16 net earnings for the year 1. 464.4 grains 2. $0.80 8. $2600 4. $3.60 5. $45.00 6. $36 7. 12 lbs. 8. $4.20 9. Milk; $1.63 Page 51 10. 4320; 103,680 14. 16,200; 388.800 15. 16,122.240,000 16. $164.26 Page 57 1. 1 in.; 9 cu. in. 2.9: 18: 45: 81 3.144 cu. in: 288: 432: 864 4. 81 6.^ inch, i, I. i. 6. ' 144; 288; 72 7. 12; 24; 24; SM; 936; 11,232; 8424; 252.720; 2106; 254,826 Page 68 10.373.33; 10,752; 233,333i 12.48 sq. in.; 336 sq. in.; 2560 sq. in.; 182,250 sq. ft.; 9x; 9x.yd; ab mi., xy 13. 27; 486 Page 58 14.81 cu. in.; 81 cu. ft.; 81 cu. yds; 81 cu. rds.; 81; 9a 15.147 cu. in.; 96 cu. in.; 06 cu. ft.; 180 cu. ft 27 cu. yd.; 180; 12x; 3xy; abo. 17. 30.000 lbs. 18,000 lbs. 18. 10,330 lbs. Page 59 19. 15,504 20. $12.48 21. 676; 108 28. 10.944 lb.s. 28. 20.736 cu. ft. 1. 6; 8; 9; 23; 22; 17; 8; 16; 9; 10; 6; 2. 2. a+b 8. a-b 4.ab; abc; xyz 6.36; 96; 54; 72; 288; 432; 1296 6.8x; 7y; 25ab; axy Page 60 7. ab; ex; 15abx 1. 8 Page 61 7; 88 9; 30 30 4 13 16 15; 14; 16 12; 13i; 20 2. 3. 4. 5. 7. 8. 9. 10. Page 68 2. 3. 4. 5. 6. 7. 8. 5 30 12 $.90 $.65 $.27 w. 14 yds, 9. 30 mi. per hr. 10. 12 gals. 11. 25 12. 84 13. 55 Page 64 2. 1760; 3520 3. 40 4. 30 6. 40 6. 70 ft. 7. 24 8. 45 9. 36 10. 40.669§ per. mo.: 1355iVoper day I I :] :{ .'4- io; 1 n. a i. ft;i: :9s 7 w, ; . h; i ;lNi)i., cu.yi. v: h: : m . 10,11 ANSWERS ik r i; ''" 61 7 7 20 11 6* Page 65 11* 37.381 12.46,693i; 51,- 923J; 41,909f; ^, 21,5441: 19,236 IS. 20 14. St. Joseph 5 Memphis Oakland Hartford Worcester Peoria „_ 16. 544; 786; 321; 17.953; 7170; 8411; 3071; 3830; 112,836 17. 8; 77; 700; 100; 10; 9; 32; 15 Page 66 f- 79S *• 243 5- 324 5- 279 I' 469 ;• 428 iJ- 742 10. 468 403 1. 2. S. 4. 6. 6. 7. 8. 9. 10. 11. 12. Page 67 2. S. 4. 6. Page 68 Page 75 i- 68 ;• 160 i' 25 "• 26.25 Page 77 1. 62,500 sq. ft. J. $2,000 *• f 1,250 * 6900 11. |1.20; 618; $60 13. $14.40; $4.80 $72; $240 Page 78 J?*v r^ *219 21.N. E. corner. 30 ft. from E. line & 70 ft. from N. hne;N.W. cor- ner 34 ft. from W. hne & 70 ft. froni N. line; «A y*' corner, 30 ft. from S. hne & 34 ft. from W. line *2. $112.60 Page 79 !• $5.60 Page 80 12,178 $162 ^ $27 $22.50 $17.28 $210 Page 81 $51.54 Page 69 6 nearly 7 3 almost 4 182.-f ^49.6 10.2+;1.58 + -w. 835.2 sq. mi. Jl. 34— nearly 130 12. almost 270 IS. 11.29 mi. 1. 2. 4. 5. 7. 9. 10. 14. 15. 16. 17. 18. 19. Page 70 147.8 , 23.08 16.7 j 13 ^$589.23 + —' ^12,908.74 20.269; 342; 235; 140; 189; 293; 15; 20.6 Page 82 1. 10Q 2.iy?;12i;llj|; •• 143ii Page 83 1901 1100 1188 3.748,\ 15271 27 Page 84 !• 27^8? *• 42? Page 84 |- 1275Jg 8.8m;^$9454.8; »• 36381?: $20,- -« 493.18; $56.32 10.6y5;$8516.23; $1229.72 Page 85 1. 7432?; $5352.- 55; $71.54. Page 86 i- 38; 38? f- 4U; .37,*. !• ?; 3?. 4. g J- 12.75 9. 78'';32<»;76* ' Page 88 1. Adams, 49| Benson Boyd Claussen Denning Doan ,„ f-.. 28Hhrs S. Adams, M $2.25;T.$2.40; W. $2.40; Th $2.55; F. $2- 70; S. $2.62i Benson, M $2.40; T. $2.55;* Page 88 ;• _ $84.45 Boyd.^'''"^^' ^- $16.20 55;' S. ',2:70' Boyd, M. 0: T «1.20; W. $1.- $2.10 ' Claussen, M $2.40; T. $2.55; W. $2.55; Th $2.55; F. $2.70; S. $2.55 Denning, M S2.55;f:$2.70;- W. $2.55; Th. $2.47i; F.$2.- 70; S. $2.70 Doan, M. $2.- 40; T. $2.40- W. $2.40; Th! $2.40; F. $2.40 |2.40; F. $2.40; ' ^ S. $2.40 4. Adams $14.92^ Benson 15.30 Boyd 8.85 Claussen 15.30 8. Adams Benson Denning Uaussen 9. Adams Benson Claussen Denning 10. Adams Benson Claussen Denning ^^Total 11: 3; 4i; 3 1* 3 (3) 4i $1.05 1.80 1.80 ^2.55 $7.20 18* Page 89 *.pr. $2875.; Cr. J1232.75; On hand $1642.25 Page 00 5. Receipts $2.00 gxpend. 1 . 25 Balance $.75 6. Receipts $105.- iS'* ..Expend. 27.61. Balance, %i29.^7r-^"«' 5- $17.50 •• $5.86 Page 91 ??• $48.75 "• J9.78 J J- $60.70 "• $36.75 Page 92 "• $2.98 1- 13c. ?• 1 3c. J- 72c. *• 4.21c. **• 35c. g:e 93 $3.47 $276.01 203.77 173.13 228.94 $24,205 29.00i 27.195 39.05 18.55 82.46 881.85 ; $2.19; J $8.96; : $8.59 $29.56 404 RATIONAL GRAMMAR SCHOOL ARITHMETIC Page M S.S153.64;S89.32 Page 96 2.0ct. S168.37 Nov. 173.16 Dec. 155.82 Jan. 166.93 Feb. 148.94 March 146.91 April 110.24 May US. 09 June 120.49 July 108.37 August 118.29 Sept. 111.51 Total $1642.12 S.Oct. $5.43 Nov. 5.77 Dec. 5.02 Jan. 5.38 Feb. 6.31 Mar. 4.73 4. $160.02 6. $166.91 : $178.- 63, Page 96 S.April $3.87 3.65 4.02 3.49 3.65 3.71 $113.83 $136.92 May June July Aug. Sept. 7. 8. 9.$575. 57; $64.92 Page 97 2.41b; y + 41b = 141b. 5. x = y 4. 20. 6. 35 6. 41b. 7. 4. 8. 4. 9. 3. 10. 8. 11. 3. Page 106 1. 10; 20; 4; 100 5. 2; 4; 5; 2^; U 3. 21; 14; 7; 2 4. *. 6. 10 6. 15; 3; 5; *; i 7. $120; 24; 12; U; *. 8. 1st; $400. 9. 50:16. Page 106 1. • 36c. 6. 696 sq. ft. Page 109 8. 64 sq. yd; 64. * 238 8q.yd. 1228 sq.ft. 9. 10. Page 109 11.2400 sq. ft.; 24. 12. 1^; 213i sq. yds. Page 110 13. 20; 30; 40; 80 14. $48,400. 16. $8,910. 16. 18 sq. ft. 17. 144 sq.ft. 18. 120,000 sq. ft. 60,000 sq. ft. 60.000 sq. ft. 19. 1080 sq. ft. Page 111 28. ^ 24. P-* ab 28. 120.000; 120,- 000; 70.000; 60.000 26. 50,000 sq.ft. 29. 60,000; 60,000; 50,000; 70,000; 60,000 80. 108 sq. ft. 31. 120.000 sq.ft. 82. 3128 sq. ft. 88. 168 sq. ft. Page 112 84. 1. 252 2. 48 3. 80 4. 4860 5. 640 6. i mn rods Pq 7. rods 8. 2y 1. 72 sq. in.;72sq. ft.; 60 sq. yds.; abc sq. yds. 2. 1360 sq. ft. S. 1088 bu Page 113 5. 355 6. 64. 7. 120. 8. 10. 9. 10. 10. 3924. 12. 567 lbs. 15. 1782 1b. 14. 180 cu. ft. 16. 1346.49 + 16. 264; U. 17. 2176; U- 18. 1080 cu. in., about i 19. 1075 cu. in.; about i; 2160 cu. in.; little over a bu. Page 114 80 lbs; 2160 81 lbs.; 2187 125; 151 4?; 27 64; 3 A 1U:1U 4,',; 31118 900; 1012^ Page 116 10. 15c. 11. 4,»,c. 12. 88c. 15. 88c. 14. 17,982 lbs.; $2174.47 16. 2,926 lbs.; $148.83 16. 146.200; $8.- 799.70 17. 8.966 tons; 73.1 tons 18. $248.04 Page 116 19. $301,436 20. $18,839 22. 110.600 bu. 1. 180* 2. 30* 5. 25* 4. 16* 6. 0* 6. - 10* 7. 15* Page 117 8. 6*; 11*; 32*; :'2*; 14*: 9*; 9*; 5i*; 3i* 9. R5*:Rl^*;R3t** I'4^*;F18*;F6*; R8f*; R21i*; R5^°; F33i*; R4ri*; R90* 10. R84*; R180*; R159*; F36*; F33*; F38*; F16*; Rlli*; F2U* 11. +69* 12. +9* 15. -7* 14. +3° 16. +4*; -1*; 0*; -8* Page 118 1. 2 2, 12 hrs. S.i; 1: A; ,S; i 4. Ihr. ^•ih't A; ^i't 9^;Vti 6. Ihr. 7. 12 8. Imin. 9. a\i of \.'hole turn. 10. 60; a second 11. 24; 168; 720; 8760 Page 119 12. 1440; 10.080; 43.200; 625,600. IS. Hourhand VI.; Min. handXII.; Sec. hand 60 14. Hour hand bet. V.andVII;mia hand bet. 48 and 49; sec. hand at 46 1. 102.400 sq. rd. 2. 640 acres; 160 acres. 5. 160; 120; 80; 80; 40; 40; 20. 9. 2600 rd. 10. 480 rd. Page 120 11. ' 6ini.: 6mi.; 12. 36 4sq. mi.; 23.- 040 14. 144 18. 160x160 Page 124 6. .02; .06; .08; .12; .25; .33i; .87i; 1. ^» A; A; «*»; Ai 8.4lb.;'9lb.; 121b. 9. 4; 12; 22; 100 10. 2in. 11. 12ft. 17. $1 18. 2in. 52in. 19. 61.2 in. tall Page 126 21. 22. 28. 24. 26. 26. 27. 28. 29. $1.25 .05; S5 $4.80 $500. 100* $9600. $50 25%; 76% 25in. 6 or /o Page 126 1. $8 2. .06 S..03; .01; .005 4. .015; .025; .035; .07; .08; .17 6. 48c; $1.50; $2.- 88; $6.10;$7.44; $27 7. $54; $81; $121.- 50; $330.75 8. $1.68; $15.75; $26.25 10. $48; $72; $120; $240; $4; $12; $8 ANSWERS 405 Pace 126 11. 6% 13. $180; $198. 14. $24.50; $70.87^ $85.75 Pace 1S8 S.Floor and Ceil- ing each 1,080. 000 lbs. ea. end wall 432.0001bs ea. side wall 540,000 lbs. 4. 44^96. lbs. 1. 39.6 mi. per hr. 2. 3.10P.M $. 378 4. 1 P. M 6. 4301 miles 6. 38 mi. per hr. 7. 23imi. per hr.; 18^. 8. 41i; 34^ Pace 129 9. 7 miles 10. 50i mi. 11. 13i 12. 6 sec. 18. i hr. 14. 679 ft. 16. 510.39 16. 2.05195 17.2712 ft.; 1329.- 45 tons Pace ISO 18. 2.82 T (omit- ting engine and tender) 10. 3.192 tons 20. 1.666 tons 21. 2.51+ tons 22. $156464.40 28. 26.66i tons 24. 11.75 tons Pace 131 1. 45i; 50i; 47; 46;49|:47i:45f 2. 11; H; i; 2*; U; U 3. A2i; B^; Ai; A2; B2; on time; Al. 6. 55.5; 24.5; 30.8; 50 6. 50min.;20; 25; 43. Pace 132 7.1.11; 1.225; 1.- 232; 1.165. 8. 1.11; 1.07; 1.16 1.20 9. 73.5; 66.9: 70.8: 72.4:66.9: 70.5; 72.8. 10. 38. 11. 1070. PaC6 188 1.88/6; $8401.56 2. $50.40 3. $63.35 4. 18f acres;' $1406.25 6. 37^ acres; $3281.25 6. *; * 7. $1500 8. 1. 9. 160; 220: 560 Pace 184 10. 3300 11. $82.46 12. 60. 13. 151.875 ounces 14. 4. If lbs. 16. $5.62^; $7.05 16. 2f; 45i; 160 17. 40^; 324; 72. Pace 186 1. ^; 2;^; 2;i;4; *;i:i;8;i;i; Aj 16. 3!EtoF= ;3;6:16 = 2;Fto E =i;EtoG=- 4; G to E -i; F to G -2; G to F -* 4. 2; k; 3 6. 3; i; 3; i 6. 1;1;1;1;1. 8. i; jV, i; iV, 3; 12; 1 Pace 186 10. 16; 12; ,S; A; 4* 32 11. V; 320; 9; i 12. 2; i;102. 400; 160; 640 18. 10; 20; |; !; i; 2; 5 2; 3; i Pace 137 1. 10c. 2. $.50 3. 8c. 4. 30c. 6. 8 ft. 6. 10 7. 12 1*8. 15; 24; 6 9. 7; 7* 10. 2; 8 ll.A = 20; C = 42i Pace 138 1. 2; i 3. 2; 4; 6; 12; 3;9; 18 4. 2; 3; 6; 12; 4;8 6. i 8. 1 Pace 138 9. 4; 2 10. ii 11.1.3; 8: t; it; 13 2. i: I: i; A; ii 3f;|:3:.H; it Pace 139 12. i; J;i;*;4;i; ♦;i; A; A; A; 13.4; i; i; A; A; 17. i; ♦; A; A; A: ii«; A; A» 1 . 1 18. l;%;%^l',l;U f;i;8; i; I; a*.; 19.^ 4 20. 6; 8; 10; 12; 14; 16; 18; 20; 22; 50 21. 1^; i)\ iJ\ 15: 18; 21; 24; 27; :W; 36; 45; *10; 90 22. 10; l.l; 20: 25; 30^ ^5; 40; 45; 50; 100: 150 23. il;a;S;t;S 24. ,%: A; ,V, ^if^; Pace 140 25. A; A; A; A;A; , ^ ill iV= «' 1. 2; I; i''« 2. 2; 3; A: A 3.*; i; i; i;?;f 4. 4; f; V, ? 6. Same 6. A: A A; U: IS; 7. 2" 2 8.'3;7;20;25;l'00 Pace 141 lO.A; \l; l«; 18; ^^2I;U;A. 14. I; I; I 18.i: i; I: i; ?; ^ Pace 142 2.1. 2, 3. 4. 6. 12 3. 12 4.'l, 3, 7, 21; 21 6. 1, 2, 3, 6; 6 6.1, 2,3, 4, 6, 12; 12; 1; 2, 7, 14; 14; 1, 2, 3, 4, 6, 12; 12 Pace 143 8. a. 1. 2, 3. 5, 7. 11. 13, 17, 19. 23, 29, 31, 37. 41. 43, 47. 53, 59. 61. 67. 71. 73. 79. 83. 89. 97; b. 4. 6. 8. 9, 10, 12, 14, 15, 16. 18, 20, 21. 22. 24, 25, 26, 27. 28. 30, 32. 33. 34. 35. 36, 38. 39. 40, 42, 44, 45. 46, 48, 49, 50. 51. 52. 54. 55. 56. 58. 60. 62, 63. 64. 65. 66. 68, 69, 70, 72, 74, 76, 77, 78. 80. 81, 82. 84, 85. 86, 88, 89. 90. fli, 92, 03. 04. 0.5. 96, 9S, 99, 100. 10. 2. 4. 0, 8, 10, 12. 14. IR, 18. 20, 22. 24. 26. 2H, :-;n, :v2, 34. 3n, ;J8, Ai), 42. 44, 46, 48, 50. 1,:T, ,1,7,9. 11. 1,^, Ifi, 17, 19. 21, 2.S, 25. 27. 2B. :\], 3.^, 35. .37, 39, 41, 43, 45, 47,40, tlA. 8; 2; J6; 38; 42; 2. 5; 9; 21; 15; 19; 27; 41; 3. 5; 2; 19; 41; 4. 8; 9: 21; 15; 26; 27; 38; 42; 12.9. 21; 8. 16; 3. 7 14.7x3; 2x2x2x3; 5x5; 3x3x3; 3x 2x5 16. 5. 1. 2x2x3x3; 2x2x2x2x3 3. 2x3x9; 3x5x7; 2x2x2x2x2x3 ; 3x3x3x2x2x2; 5x5x3; 5x5x5x2 Pace 144 8. 12; 14: 12 9. 1. 20; 2. 18: 3. 45; 4. 215; 5.36; 6. 22; 7. 63; 8. 163 10. 1. g; 2. 2S; 3. ."n; 4. S; 5. ,•; 6. ih 7. n; 8. .?,; 9. I; 10. ?; 11. A; 12. 4 406 RATIONAL GRAMMAR SCHOOL ARITHMETIC 1. 2. S. 4. 6. 6. 7. 8. 9. 10. 11. 12. 1 Page 145 Page 146 1^: .. a: it; I'o'b Page 147 i: I *./o;l:a; A; A; . A; W, 85; Si 6J3 ^6^ 70 60 12 2 20 120; 8. Uo"; Page 149 2! 6. 7. 8. 11. 40; 24; 75 360 Page 161 2. 36; 120; 180; 1050; 168; 108 4. g§i; igg; AV, Page 164 8^8 25,='i^ 109iS Page 166 William i ft. 6. 518; 85S; i3/o; SOU; 413*3; 227 8. 23iS 9. 21 Page 166 4. 6. 6. 7. 2. 3. 4. Page 166 14. ^la\, 16! 65f; $1*1'. 7^ Ji- ll: '^^ 1 Page 167 $21. OH 300; j\ 32 32 V fi 20. 21. 22. 28. 24. 1. 2. 8. 4. 6. ?: n 8. 12; 4; 3; 6; 5; 15; 4; 8; 6; 6; 10; 10 Page 168 9. 21; 2j; Ig; 7i; 3; f ; 3*; 2f ; 5i; 31 ; 12^; 8i 11. 5; 7; 7; 5; 19; 16 Page 169 1. 2. 3. 4. 6. 6. 7. 8. 9. 10. 11. 12. 13. 14. 3| 231 5i 7\ 6* $7 16. 3f; $1.20; $6.00 16. 5i 17. li Page 160 1^6; 20; 28; 10; 12 18; 20; 16; 29}? 2. 2Si; 231; 89i; 231?; 233i: 189i Page 161 1. $7 2. $70; $5 3. 154; 192i 4. 977* 6. 357 Page 162 1. 15 sq. in.; 48; 56 sq.ft. 160 sq. rda.; 320 sq. rds ab. Page 162 2. 4 sq. ft.; 14 sq. ft.; 27 3. V»5 sq. in. 4. i sq. in.; A; *; A; A; A; §i; ii; Si Page 163 Page 164 8. 1277A t: «^l 6. 1390^9 7. 6I3JJ 8. $4.12* 9. $5,841 10. $10,223 11. $2,931^, 12. $9354^9 Page 166 1.4; 5; 14; 16; 5; 12; 8; c; m 2.18; 32; 108; 325 723; 14,665 1- '^ i S 6. 25c. Page 167 5: . '^ 9. ^e; $5,000 'I. % Page 168 8.135; MS; hV, Mli; m; 2?a; 9A; Hi; 2A; 8i 4. 12*; 18.67?; 16i; 12.67i1, Page 171 1. $3i 2. , i 8. 39113 lbs 4. 193 6, 66 6. 4f 7. ; or 3* 8. llg 9. 3J 10. 41 11. 3* 12. 10 13.6g; 9?; 3|; 8V, m 14.2U; 143; 9g; 8 " 16.9!'9J; 7*; 9A; 6.^ J Page 172 20. V 21. $3100 22. 1935 23. 105 Page 173 J- ^* 2- tSi 8. 2Ji J- if 6. la ?: -^ 8. 9. 10. 11- 4^ J8. l.VA !*• li\iW 16. ^ be 1. 150 lbs. 2. R. 11 lbs. Page 174 8. R 3x 4. 4 m lb. 6. R 6* 6.R 35X; R 51; R 5115; L 45,»«; L 65A; L 597A 8:B 44,',; F 213^, TB7/4: F9/" Bl|8;B5X 10.D4i4; U14S;D 781; D 64=^ oa. Page 176 12. + 3tlb8.; +ioz + 33* T; +205- ilbs l.V/; tS«; i8: tkl; m; lAVo; 121?; BU; 624; ia§; A; ir%; . 2|}V, a%; iig 8.A; i4; 25; 1,VA %;3»A;A;iU; m; 141; O; 10019; 10; 26}?; 71ii Page 183 1.4* ft.; 3f ft.; f>'; 2r;2|';U'; 11' 8.20'; 12*'; lo'; 3*'; 7*'; 3i';6i' 4. 298.'«sq. ft. 6.75'; 33"; 60*; 2r 7.33'x75«'; 30*'x7r 6* above center and at sides; 9*below ANSWERS 407 Page 184 9. 32*; 48": 48" 11. 9.492 sq ft. 12.24" each; 28";2(r 14.84"; 60"; 114"; 84"; 9" and 3" 16. 11; 10: 4^ Page 186 1.320'; 400'; 470- 156 sq. rds. 2.200; 80; 58(V«»i 5. 200; 160; 4. 266i 6.56'; 80'; 40'; 20' 6. 50'; 85'; 80'; 90'; 110'; 120'; 170'; 210' 7. 60'; 30': 15' 8. 70,«oV9;i3,y,: 6.61885: 20AV.; lll/o% 10. 41rgcg sq. rds. Page 186 11. 13012 rds. Page 187 1.7*; 6; 18; 49; 4; 2b; 6; 6; 2ab; a 2.32; 15; 72; 88; 4?; 24; 38; 2; 2; 15; 25; 2; 2; i; i: 3i Page 188 1. 75c. 2. 60c. 3. 90c. 4. 12 6. 60c. 6. 30 7. 2.40 8. 18 9.649; 663; 38.50 10. 24; 32; 34 11. 665. 96«, 12. 28 13. 29 14. 2 ', 16. 7607i Page 189 16. 272,160 17. 49 19. 240; 50; 40 20. 30 ft. 21. 40 ft. 22. 108 22. 108 23. 31 24.9^; or; 3i";22i" 26. 9" to mile 26.1" to Sift.;!" to 25': V' to Of Page 190 27. Water, lOiozs.; Fat, 82; Prot., 71; 6?; Water. 18/oOzs.; Fat 151; Prot., 12i'<»; 12,V; Water, 40JOZ8.; Fat, 338; Prot. 31 i; 261 28. Water. 681ozs.; Fat, 201; Prot., 2U; 17r'.: Wa- ter, 206|oz8.. Fat. 62?; Prot., 64; 51 i; Water, 258 ozs.; Fat. 78; Prot.,80; 64 29. Water, 15iozs., Fat, 5i; Prot., 5: 4; Water. 73 ozs.; Fat. 2t; Prot., 2i; 2; Water, 9/nOZ8.; Fat, 3/o; Prot.. 2'g; 2i; Water. 25iozs.; Fat, 8-: Prot., 8; 63 30. Water, 4iozs.; Fat, 2,',: Prot.. l/o; 2i; Water, lOgozs.; Fj Fat, 5i; Prot.,4i;6i; Water, 6{t; Fat, 3?i; Prot.. 2i|; 4; Water, 32iozs.; Fat, 16: Prot.. 12i; I8i Page 193 2. 36.04 3.1891 25.54 in. 1892 36.56 1893 27.47 1894 27.47 1895 32.89 1896 33.14 1897 25.85 1898 33.77 1899 31.49 1900 28.65 1901 24.52 Total 327 . 35 4.Jan.21.89 1.99 Feb.25.15 2.88 Mar.26.50 2.41 Apr. 18.75 1.70 May33.48 3.04 June39.25 3.57 July33 38 3.03 Aug.29.04 2.64 Sept.31.38 2.85 Oct.14.54 1.32 Nov.28.50 2.59 Dec.25.49 2.32 Total, 327.35 1. 223. 76T 2 157. 42i 3. 127.35 4. 1095i Page 193 159 792 1829.1 Page 194 2. 16.875 3. . 133 lbs. 4. .08171b. 6.Wt. of water. .09751b.; org- anic .031 6. Poplar water. .089; organic .0387; Ash wa- ter, .055; or- ganic .0414 7. 1.0771bs Page 196 8. 50.625 cu. in. 1.04 1.48 .04 2.71 .06 3.51 .08 2.85 .10 3.20 .12 3.16 .15 7.36 .26 12.13 .31 15.04 .33 17.16 .34 20.56 .34 18.61 .45 16.10 .41 16.31 .35 17.67 .35 18.74 .35 18.73 2.. 44 .44 .40 .38 .56 .54 .36 .34 .36 .34 .35 .32 .29 .18 .08 .03 .05 .03 .03 .02 .0 -.03 -.14 -.11 -.07 -.06 -.03 -.03 .0 Page . 6.37 7.42 11.15 9.82 16.04 20.59 20.16 11.03 4.87 7.92 .04 -.46 -3.89 -4.80 -3.73 197 5.14 6.62 11.81 9.47 16.08 16.39 15.39 8.12 2.75 4.52 . 1.99 2.05 -4.10 -6.16 -4.80 •f.Ol Page 197 4.2yrs.; 17; bet. 40 and 50; bet. 40 and 50 6.2yrs.; 13; bet. 40 and 50; bet. 50 and 60 7. Boys Girls .036 .040 .052 -.020 -.020 .080 -.033 -.003 .043 -.046 .010 .039 -.135 .053 .132 .013 .115 .020 .549 .171 Page 198 . 1.3.811,808; 381.- 180.8; 38.118.- 08; 3,811.808; 38.11808:3.811. 808 2.1; 2; 3; 2; 4; 4; 4 3. 0; 1; 2; 3; 5; 3 Page 199 8. 143.547 9. 806.7525 1. Total wgt.. 3236; H. puUiu lbs., 242.7; tot. wgt.. 3377.8; H. pull in lbs., 253.335; total wgt.. 5256.35; H. pull in lbs., 394.226i; total wgt., 6258: H. pull in lbs.,469.- 35 2. Total wgt., 2240; H. pulfin lbs., 280; total wgt., 1709.58; H. pull in lbs., 213.691; total wgt.. 5054.85; H. pull in lbs. 631.856i; total w?t., 5335.75; H. pull in lbs., 716.961 Page 200 8. 1060.9. 273.- 7122; 4446.38. 1147.16604 ; 6669.25, 1720.- 6665 408 RATIONAL GRAMMAR SCHOOL ARITHMETIC Page SOO 4.(1.) 7855.65, 223.8860i; 149.- 25735, 2ttl.- 593145 (2.) 11530.86, 328.62951; 219.- 08634, 383.- 977638 (3.) 12600.65 359.118525: 239.41235,419.- 601645 1. $62.75 2. $62.75 Page 201 8. $37.25 4.(1.) 18011.444 18735.556 62795.833 67801.667 106885.636 (2.) 274230.136 5. 1. 14.29 2. 14.43 3. 14.48 4. 14.43 5. 14.35 6. 14.32 7. 14.45 8. 14.22 9. 14.33 10. 15.009 Page 202 6..1; .01; .001 6. .5; .05 7.. 10; 1; ,',; ^h^ 8.. 01; .1; .01; .001 9.6.8; .68; .068; 68.; 680; 6.8; Page 203 11.1. 23.8 2. 2.38 3. .00238 4. 2380. 5. 23800. 6. 238. 7. 2.38 8. 23.8 9. 23.8 1.8122, 473, 954.- 50 2.46,073; 1355a\ 8. $6,483 4. 8298.70 6. 88.785 6. 823.126 7. 8601.276 Page 204 8. 240.1; 201.9; 442.10 9. 812.85 1. 3.144 2. 3.138 Page 204 3.3.142; 3.151; 3.142; 3.160; 3.141; 3.193 av. cir., 30.809; av. dia., 9.77 ratio. 3.1546 Page 205 4. 3.119; 3.143; 3.142; 3.142; 3.142 5.Av. cir., 134.3; av. dia.. 42.18; ratio. 3.139 7.3.145; +.0034 diff. Page 207 1.18. 3.69 19. 3.69 20. 3.78 21. 3.82 22. 3.88 23. 3.87 24. 3.92 25. 3.92 26. 3.88 2.18. 1.87 19. 1.85 20. 1.86 21. 1.85 22. 1.87 23. 2.04 24. 2.11 25. 2.10 26. 2.07 8.18. 1.97 19. 1.99 20. 2.04 21. 2.06 22. 2.07 23. 2.04 24. 2.11 25. 2.10 26. 2.07 l.Alum 2.664 Zinc 6.984 Oast Iron 7.2 Tin 7.333 Wrt.Iron 7.68 Steel 7.840 Brass 8.381 Copper 8.832 Silver 10.481 Lead 11.350 Gold 19.214 Plat. 21.552 Wood Cork .24 Spruce . 5 Pine.Yel. .554 Cedar .561 Pine,Wht. .448 Walnut .670 Maple . 750 Ash .844 Beech . 852 Oak 1.04 Ebony 1.216 Page 207 l.LignumV. 1.338 Liquid Alcohol . 800 Turpen. .870 Petr. .891 Olive Oil .912 Lin. Oil .952 8. Water 1.026 Milk 1.032 Acetic ac. 1 . 064 Mur. acid 1 . 2 Nitric ac. 1 . 52 Sulph.ac. 1.842 Mercury 14.08 Page 208 2. Gas Hydrogen .069 Smoke (wd.) .090 Smoke (B.C.) .101 Smoke, 212° .469 Carb.ox. .967 Nitrogen . 974 Air 1. Oxygen 1 . 105 Carb.ac. 1.529 Chlorine 2.440 8.Glass(av.) 2.81 Chalk 2.79 Marble 2.71 Granite 2.66 Stone (com.) 2.53 Salt (com.) 2. 13 Soil (com.) 1.99 Clay 1 . 95 Brick 1.89 Sand 1.82 Page 209 2. .75; .40; .80; .875; .625; .187; .156; .203 3. 1.50; 3.20; 2.- 75; 6.875; 13.- 312; 18.406 4. . 14285; .23076; .47058; .39130; .43243; .26315 6..333333;.666666 .166666; .833- 333; .111111; .222222; .777- 777; .363636; .545454; .8181- 81; .373737; .4- 07407; .740740 6. 1.35; 16.875; 5.286; 17.9625; 20.04; 1.008; .00875; .0875; .000625; 30.06- 060 Page 209 7. .9166: .7692: .3684; .2727; 4.2222; .4666; .8589; .7107: .5947 8. .4702 + 9. .301 Page 210 1. 491.07^ 122 . 767 1257f 11314? Page 211 2. 1413.72 8. 268.91074' 4. 21480.945 lbs. 6. Diam. 1.7506: area, 2.407 sq. in .' 6. 7976. 71875 lbs. 7.1. 7951.4956 lb. 2. 47078.996 lb. 3. 31175.346 lb. 4. 37192.960 lb. 5. 42270.6 lb. 6. 31569.56 7. 73378.313 8. a- ^ xr 2 9. C = 2HR 10. 763.41 sq. ft. Page 212 1.1. 99.89 2. 317.702 3. 1081.4 4. 366.731 5. 2917.7183 2.1. 78.58 2. 97.04 3. 12.93 4. 36.159 5. 37.896 6. 801.32 7. 723.037 8. 2.797 9. 25.252 10. .017 11. 58.1211 12. .0003 3.1. 3759.52 2. 677.8582 3. 756.0743 4. 8680.0868 5. 71.517 6. .71517 7. .06464 8. 1.209(5 9. 16.210801 10. 12.096 11. 12.093 12. 79.105488 13. 79.105488 14. 79105.488 15. 2184.1974 16. . 21841974 17. 52.904544 18. 62.904544 ANSWERS 409 Page 212 4. 1. 4.5009 2. 4.591 3 88 4! 320.152 5. 28.4 6. 38.6 7. 4620. 8. .0462 9. 462. 10. 462. 11. 462. 12. 231.000 5. 1. 26.6661 2. 876.388.888<i 3. 1285.714 4. 122222.22 5. .333i 6. .333i 7. .666} 8. .444j( 9. 4888.888 10. 4.333i 11. .855K 12. 12.312 13. 3636.363 14. 7575.757 16. 78.787 16. 7.878 17. 3.463 18. 11.441 19. .191 20. .636 21. 112.222 22. 33.333i 23. 186 24. I.IIU Page 220 1. 230c.; 2300; 23000 2. $2.48; $24.80 2480 S. 49: 4.8 4. 10 pounds; 200 marks; 250 francs; 66f rubles; 250 crowns; 250 liras 5.20.54 pounds; 419.29 marks; 518.13 francs; 129.52 rubles; 492.61 crowns; 518.13 francs.. e, $7.9625 ' 7. $323.08 1. (1) 288; (2) 1440; (3)4080; (4) 4080; (5) 4608; (6) 26880 2. 10 5. 30 4.1330 grains; 2il troy ozs. 6. 128 ounces; 96 6. 384; 774 7.4800; 22400; 8008 Pace 220 8. (1) 56. (2) 300. (3) 1391. (4) 1728, (5) 13,008, (6) 15,200 9. (1) 2 tons; (2) l\ tons; (3) 8 tons; (4) 4i tons; (5) 64 tons; (6) 6* tons Page 221 10. (1) 3 long tons (2) 8i 1. 1. (3) 5 1. t. (4) 42i 1. t. (5) Vfl 1. 1. (6) If 1. t. Il,5001b8.; I1201bs 1. 198 inches; 792 inches 2. 66 ft. 3. 678 inches; 466 inches 4. 32 rods. 2yd8. 6. 60ft. 6. -237.6 hds. 7. 2840.2 bds. 8. 90 9. 336/r 1. (1) 432 sq. in. (2) 440; (3) 1.296; (4) 5.616; (5) 46.704; (6) 6 588; (7) 4.636; (8) 6.516; (9) 39«204 2. (l)4sq.rds. (2) 4 A. (3^ 6400 sq. rds. (4) i A. (5) 272i sq. ft. ^6) 4840 sq. yds. 8. (1) 1 A. (2) 200 sq. ch. (3) 6400 sq. ch. (4) 230- 40A. (5) 160 sq. rds. (6) 102,400 Bq. rds. 4. 2rdB.; 10 rds.; 40rds.; 160 rds. , Page 222 S.lOrds; 36sq rds more ' 6. 60 sq. rds more 7. 79 sq. yds less 8. 43.560 9. 201 sq.yds more 1.1. 5pts. 2. 10 gills 3. 30 gills 4. 16 pts. 5. 48 qts. 6. 94 pts. 2.1. 12 qts. 2. 6 qts. 3. 9^ gals. 4. 2 gals. 5. 8qts.,lpt. 6. 10 gal8.,2qts. Page 222 3. 57^cu in.; 2S| cu. in. 4. 37.8 cu. in 6.1. 12pk8. 2. 9pks. 3. lOpks. 4. 32dry gals 5. 2ibu. 6. 320pts. 6. ,\|lb. 7. 28bu.; 25bu. 8. 806ibu. 32.256 tons 9.About i Win- chester bushels. 10. 224pt8.;2ibu. 11.31.08 per bu.; 3|c. per qt. Page 228 l.I. 3600 sec. 2. 86,400 sec. 3. 604,800 4. ,iohr. 5. 2hr8. 6. 7da. 7. 2wka. 8. Iwk. Ihr. 5 min. 52 sec. 9. lO.OSOmin. 2. 52 wks. Wed. 6. $18171 6. $2184 9. 12:37hr8l9min 12:66hr8 17 min 13:13hrs 16 min 13:29hrs 16min 13:45 hrs Page 224 10.6:14hr8.; 6:23 hrs.; 9 min. 11.6:26; 6:30; 4 min.; 6:36; 6:37 Imin.; 6:46; 6:- 43; 3 min.; 6:45 6:50; 5 min. 13. Jan. Ist. 1:39 April 1st. 1;35 July 1st. 2:5 Oct. 1st. 1:30 14. Jan. Ist. 1:39 April 1st. 1:34 July Ist. 2:5 Oct. Ist. 1:30 1. $4.20 2. 70; 2000 years; 24000 months. 8. 1776 4. $24 6. 129.60 6. $432 1. 15552 2. 46A^ Page 226 3. $25 4. $37.56 6. 1,244.160 6. 2804 Page 226 7. 7rms, 7 quires; 61rms., 9 qrs 8. 560 shillings; 6702; 26.880 9. 216; 864 10. 27,783 11. 82.266 12. £119. 6 s. 11 d. 18. 234; 2808 14. 2838 16. 78yd8.; 2ft.;6in 16. 19.200; 160 17. 19,371 18. 12cwt, lOlbs Page 226 19. 5781in 20.160yds. 1ft. 9in 21. 277qts 22. 55815 28. 635 24. $1925.92 26. 37U 26. 116.1 27. 54.87 copper; 18.29 nickel 28. 232.2 ;;. 18H so. t; I: ^; 1 81. 8.616 o8 82. 1.324- 88. 36 pounds; 700 M 84. 283.8 86. I? 36. 54ft. lOln. 8/. 2mi., 44ch., Ird 4li. Page 227 88. 112rds., 4ydfl.. lOJin. 89. 24.2 40. 660 41. 19881 42. $975 43. 5AA 44. 10,000 46. $192.50 46. $14,400 47. 6392.208sq rds 48. 27.720; 16,'i 49. 10. 60. .37i 61. $4.61 62. $45 Page 228 63. 31.600 64. 7.92 66. $122828.28 66. $530.30 67. 24589.2 68. $22.76 69. $1.80 60. 132^cu. in. 61. $727.27 62. 1188.46 63. $2117.5 64. $992.30 410 RATIONAL GRAMMAR SCHOOL ARITHMETIC Pace 228 66. lpt.a.376gill 67. $13. S6 68. 15S3 Page 229 69. Gain, 6.24 71. .921875 72. 478.12 73. $76.65 74. e'« 75. 13ic. 76. 2150.22 77. 10.8 78. $116.40 Page 23S 2.6m; 8dm; 7cm; 5mm 6. 2.112km 7. $.711 8. 48. 9. 51 . 52m 10. 4035m 11. 10,000; 1,000- 000. 17. 215.28 sq. ft. Page 234 20. 42m^ 21. 262. 12m2 22. $1664 23. 372 ares 24. 35m; 26. 258.7846 26. 2000; 2D1; .02 Kl; .6804 qto Kl; .6804 qt. (liquid) 30. 204.6211b 31. 1000 32. 13 Jibs.; 6000g. 33. 39.81456 34. 220.46; 1102.31 35. 38.58 capsu 36. 32.15 cents 37. $39.41 38. $17 39. 56 Page 235 40. 31.25 41. 131. 7ei 1726 Page 236 1. 24 Page 237 2. 60% 3. 40% 4.. 834; .62i; .911; .14?; .71f .44*; .63/ .46^',: .485;; Page 237 5. 481!%. 7.334%; 25%.; r.334%; 25' 4lf%; 77r 97i%; 21% 7iP!^ ; I. 8.1. 2. 3. 4. 6. 6. 7. 8. 9. 9. A; ,\; 10.. 07; .0625; .R;^ .875; .015; .0275; .STj; .001; .0001 11. 11.27CU. hi 12. 4% Page 238 13. 64.7; 59.2: 58.0; 64.2: 5U.0 38; 37.9; 37 4 14.25%;.12i;,[Xil .18i; 18i; .OrtJ 15. I9arx» 16. tH\ 17. $2000 18. $21W> Page 239 1. $4;$7 .^n 2. .02a; .OJt.; .02x; .12x 3. .05a; ^or^h, .05x; .05z; .40/. 4. .125a; .12o%; 1.25X 6. .Oa; Ox; .OOs: .12y; .45z 7. .16; .29; n^i; ox; om; . J*'n 8. 1.6r;3.5r;()f3i; .OrR; f'rr, .02rar 9. p^ I'^r Page 240 SJU 1. 2. 2av. 3. Vi(^. 4. $ 80 5. IfiO 6. $128; $179.20: $224; $40; im; $560 7. $,^KJ2 8. ' ■$440'> 9. SOhf Page 241 10. 1775% 12: 27 A % 13. 66§% 14. 11.200 15. $52.80 16. $14.00 17. 20% 19. 32.000 20. 697 21. 2000 22. $9,775 23. $750 24. $250 26. $29.76 26. $1 . 16i Page 242 27. $5,000 28. 15% spilled; $4.5 29. $925 30. 20%, 1. 233't; 483?; 22;f; 293't 2. Feb. 35?; 39?; 25; 324 Mar. 38g?; 45^; 16,^; 35Jt Apr. 33J; 43i; 23i; 43i 3. 213?; 24!?; 29H: 24i? 4. Cloudy. 28J5; 20IS; 261!; 242S P. Cloudy, 261!; 261!; 19il,; 23i! Rainy or S.21?; 21?; 23,S; 30?? 5.Jan. 71?% Feb. 68)%, March 81?% Apr. 100% May 77»% June 64if% July 80% Aug. 67;%, Sept. 74?%, Oct. 82;% Nov. 74?%, Dec. 82;%, 6.Jan. 82%; Feb. 66% Mar. 90% Apr. 83% May 80% June 70% July 73% Aug. 67%, Sept. 76% Oct 84.9 or 85% Nov57.7or 58% Page 243 7.Mt.T. 88.9% BIk. Isl. 94.9% Chicago 100% Cleve. 112.9% N.Y. 114% Buffalo 1 16% Boston 147% Phila. 152%, St.L. 171.7% New Or. 195%, Louis. 206% Wash. . .228% Roseb. 476.4% 8.Mt.T. 34.6% Blk. Isl. 3165% Chicago 100% Cleve. 108% N.V. 2fi2% Buff. 1.07% Boston 658.4% Phila. 703% St. L. 145% N. Or. 1613.7% Louis. 156.7% Wash. 734.8% Roseb. 158.8% 9. 39.5%; 2.4% 10. 23: 1.6 11. 120 sq. ft. Page 244 12.Jan. def. 46.8% Feb. ex. 43.6% Mar. def. 33.7% Apr. ex. 38 % 1. 60.2% 2.St. Louis. 9 hrs. 29 min.. and 51 . 72 sec. St. Paul. 8hr.s.. 49 min.. and 14.16 sec. 3.St. Louis. 14hrs 51 min.. and 52.14 sec. St. Paul. 15hra.. 31 min., and 2.52 sec. Page 245 hrs min pcc 5.F. 9 52 55.2 M. 11 11 25.62 A. 12 36 31.32 M. 13 57 46.44 J. 14 55 58.08 J. 14 57 54.9 A. 14 28 52. 2S S. 13 13 51.24 O. 11 49 18.48 N. 12 7 25.50 D. g 23 16.44 3. 6.5%; 89% 4. 27.6%; 5.2% ANSWERS 411 Pace 246 7. 21% 9.L.Pur 142.8% Florida 2.9% Texas 17.9% Mex pur 21.4% Tex. pur 3.2% Gads, pur 1.4% Alaska 19.02% Hawaii . 18% P.Rico .09% Phil Is. 3.07% Guam .006% Is. of P. .02% 10. 73.4%; 26.6% Page 247 U.N.A. 43.1% S.A. 66.4% Asia 44.7% Africa 83.5% Europe 38.5% Aua. 77.1% 12.N.A. 725 S. A. 336i*r Asia 79^S Africa 438? Europe 350 Austral. 350 13. 48.7%; 72.5%; 69%; 98.5% 16. 7.7+; 2.2; 1.- 7+; 1.4 + 16. 47.6; 33i; 47.6; 47.6 Page 248 17. 92.8%; 104% 19. 71.9% 20. 96.6%; 86.8% 1. 64.00 2. 162.50 8. 61. 50 4.1. 6100 2. 6382.50 3. 63.00 4. 614.25 5. 63 6. 6112.50 7. 675 8. 6240 Page 249 6.646.25; 664.75; 6231.25 6. 62706.12 7. 61119.52 8. 613.684 9. 653.6085 10. 617.523 11. 63886.40 Page 260 2. 660 3. 62992.50 note 62975 4. 65301 Page 260 6.1. 630.375 2. 6128.25 3. 61054.62 4. 41.2965 6. 233.284 6. 6686 7. 617.859 6. 24% Page 261 1.1. 2. 3. 4. 5. 2.1. 2. ao 3. Jlf2; 4. io 5. ^; 6. has 7. ^^ mos 3.1. Hi!; 2. 429 41U 28r* 46 26i' on . m s' hha. -3.. rho rns 1; 4. has; ;6. zus ais umo 4.1. oo; 2. ee; 3. aah; 4. ren; 5. Z2s; 6. hhrr 5.1. mo; 2. io; 3. hao; 4. hzr; 5. rio; 6. mao e.l.k k; 2.ss; 3. lab; 4. aoh; 5. rre; 6. bba lab; 4. aoh; bbaa; 1. ck; 2. ok; 3. blk; 4. bra; 6.aok; 6.cck 8.1. uu ; r zs; .3. hns; rrr ; 6. Page 263 1.631.50:631.50; 663; 6157.50; 6110.25; 6149.- 625; 631. 50t 2. 6100; 699.20: 6297.60; 6678.- 661; 699. 20x 3. 9128 Page 264 6. 662.30 6. 6332.4375 7.1. 634.125 Amt. 6734.126 2. 633.90 Amt. 6433.90 3. 631.43 Amt. 6242.43 4. 620.504 Amt. 6170.504 5. 619.565 Amt. 6299.565 6. 619.40 Amt. 6379.40 7. 620.75 Amt. 6280.76 8. 668. 44$ Amt. 6568. 44$ 9. 639.25 Amt. 6339.26 10. 6142.50 Amt. 6767.50 8. 6100. 9. 6150. 10. 670.44 11. 6240. 12. 6860. Page 266 16.1. P = 6200 A = 6219.50 2. P = 6640.22 A = 6809.24 3. P = 6720.00 1 = 6278.50 4. P = 6135 1 = 663.42 5. P = 61200. 98 A = 61323. 48 6 P = 63424.864 A = 63600.86 7. P = 62888.88 A = 2921.387 16. 34?% 17. Il3h% 18. 12% 19. S9^i 21. (1) 5.6 + ; (2) 10.1: (3) 11.9; (4) 10.4 22. 4; yrs.; 3 mo. 15 da. Page 266 23. 5 yrs., 9 mo. 24. 1 yr., 6 mo., 15+ days 26. 12 yrs & 6mo. 26. 16fyrs.; 14?yr8 27. 6yrs, Imo., 16^+ a», Page 256 30.1.6yr8, 2mo, 8^ or 9 da. 2.1yr, Imo, 15 da. 3.3yrs, lOmo., 27dfa. 4.3yrs, 4mo, 18 da. 6.3yrs, 3mo, 6 da. 31.1.1,611.78; A = 671.78 2.1, 641.927; A = 6216.927 3.T, 2yr.,3mo; 1=6126 4.T, 3yrs, 7mo. 15da; A = 6617. 50 6.T, lyr, 7mo, 26da; 1 = 83.85 6.P, 61500; A = 2009.25 7.1, 687.161 8.T, Syr, 7mo, 6da; A = 6937.- 986 9.R, 8.9% Page 267 1.1. 637.50 2. 675.00 3. 6243.75 4. 6137.60 6. 6968.75 6. 637. 60x 7. 37.50t 2.1. 662.60; 2. 6206.25:3.6525 4.6250; 5.6518.- 75: 6.637. 50t.; 7.612.50rt.; 8. 612.50 ru; 9. 612.50 ru. Page 260 4.1. Int., 642.04; Amt., 6292.04 2.1nt, 6162.365; Amt, 6797.365 3.1nt., 6537.53; Amt., 62937.53 4.1nt., 6330.46: Amt., 64195.46 5.1nt., 6771.83: Amt., 64411.83 1. 6109.09 2. 6132:622.91 Page 261 8. 666.79 4. 661.47 9f H547I W.40 412 RATIONAL GRAMMAR SCHOOL ARITHMETIC Page 261 6.1. D. $5.90; Pro. $59.41 2. D, $31.21; Pro., $284 3. D. $46.70; Pro., $462.36 4. D, $123.12; Pro., $730.88 5. D, $229.77; Pro., $810.42 6. D, $376.48; Pro., $1167.64 7. D, $360.31; Pro.. $1862.19 8. D, $193.99; Pr... $3920. 137 9. D, $157,438; Pro., $3920.137 10. D,$1101.876 Pro., $875687 Pace 263 2. $460,676 3. $1920.669 Page 266 1. 4 2. 9'5'' 3.1. 85*' 2. 61 r 3. 87' 4. 102* 5. 137' 6. 65' 7. 57'5r 8. 27'lir 4. 102* to 1' 6. 631' 6. 68ft. fin. 7. 140iinches 8. 191.2 inches 9. 127Vinches; 89i inches; 140i inches 10. 25i X 25i 11. 3'; 49*; 27' Page 266 12. 18'; ll3f 13. RI9'; C119? 14. 264; 169? 15. 60mi an hr 16. 2i times 17. 181535 lbs. 18. 121.333*lbs. 19. l,080,868i lbs., or 540.434iton8 20. 2522.026^1b8. 21. 2972.387H lbs. Page 267 22.La422.7tVil.;lb8. 2.3H7:*.]n V:^lbs. ^{,4H2hV47:M lbs. 4.4773,8:lH->-:-.:^bs. ti.r»f574.5.i8^ lbs. 7AM2'lM'2Uf: iba. 8.r>-')7'i .'■<- ^bs. 9,711;; ■■ I' ! 13 'bs. lU 792«. 31375 12. 8370.T2ai', Page 269 1.1. 2. 3. 4. 5. 2.1. 2. 3. 4. 5. 3.1. 2. 3. 4. 5. 5.1. 2. 3. 4. 5. 2161b8. 3201b8. • 6621h8. 817lb8. 1223ilb8. 85lbs. 2031b8. 2701bs. 4801bs. 572*lb8. 97.501b8. 1551b8. 1171b8. 4251bs. 212.51b8. 10401bs. 12801bs. 26501b8. 38801b8. 45001bs. Page 270 9. 1. 20; 2. 20; 3. 3; 4. 40; 5. 7000 6. 40^1 + ; 7, 60; 8. 12 10.1. 1785.71lb8; 2. 4821.42lb8.: 3. 2010lbs.; 4. 5910.711bs.; 5. 2449.291b8.; 6. 39410.71b8. Page 271 11. L = 35.71F 2. L = 66iF 3. L = 30.30F 4. L = 52.63F 5. L = 83 33iF 6. L = 52.63F 7. L=125F 8. L = 100F 12.1. 9998.801bs. 2. 5177.95lbs. 3. 2806.806lbs.; 4. 6277.818lbs. 4. 6277.8181bs; 5. 4999.401b8. 6. 24,997lb8. 13.1. 1999.801bs. 2. 10.998. 90Ibs: 3. 3999.60lbs. 4. 15,998. 401b8; 5. 799.921b8. 6. 1599.841b8. 14.1. lO.OOOlbs. 2. 50.0001b8. 3. 60,000lb8. 4. 40.0001bs. 5. 70.0001b8. 6. 72401b8. 15. (1) 200; (2) 300; (3) 149.69; (4) 90; (5) 180; (6) 54; (7) 1620; (8) 32.4; (9) 1100 Page 271 17.1. JsjHjnibs.; 2. •^l,*Mni\h:.\ 3. ^UMfMi^lliti.; 4. ai^HKl; 5. 2i^JiMii; 6. :i2Ai>iK 7. j:V-^N(h 8. ^5,fjS0' 9. .u).:^m 19.1. lU,'j.{>00tbH.' 2. ll4,00Cthft.: 3. CMJ.OOOtba.; 4. 122,4001be.; 6. I41.000lbs; 6. 14S.e5fJ 20.1. 443:£ 2. 232ft, 80 3. 4.flHfi 4. 9,0025 6. . -inSfi 6. J. 2021 5 Page 29& 18. 66S24 t<n. ft, 19. 741 SCI. rr. 20. 723 »<i. rr, 21. 172,^^ at I. rii^ 22.168/i\ sq. rti^.; 1631J? 23. 4U\9fi.[i. 24. lAA 25. 2^Ae 4|fA 26. S|A 27. 8A; 0l7f aq.rd^. Pagi 29a 1.1. r,frw- 2. 90* W; WSi 3. 30* K: ;<(rN; 4. 90'W. ;i,=^°:v I 2. l. + 7/t°. +40^i 2. +7^l^ +40°; 3. +7fr\ +4.i^^ 4. +90^. +^45^; 5. +105°, +40: 6. +30°, +»0; 7. A'h f 52* 8. -y, +50" 9. +15^ +52^ 10. +2rf, +80" 11. +68", -55" 12. -5^. +3^^^ 3. 65°; 72*; 30^i 4. 46°; 70^: 50^ 5. Hfr^ SO" 6. 36*; 22*: :<7* 7. 35'*: 1 5": 30" 8. 30"; 60°: 45° Page 399 9. 45" 10. 15^ 11. 16"; :45^ (iri*^; near AH"" 12. 30°; uear 50*; 15* 13. 15**; npar 35''; 15*; 30" Page 299 14. about same; same; 10*; 7* 16. 16.48; 35.42 16. dif. in dif. in lat. long. 1 2.28 262.01 2 70.78 50.80 3 21.53 91.09 4 6.05 63.04 5 93.26 36.52 6 5.67 200.62 7 42.00 7.89 8 20.83 18.58 9 37.80 8.20 Page 800 2. 7o'c;12o'c 8. 60*,46' Page 801 4. 7hr., 3miTi. 5.1 St. is east, 45*, 30' 6. 1st. from E., 2nd. from W. 7. 20*15'; 92* 10. 64*33'; 54' 11. 166°23'24'; llhrs. 6min., 33.68ec. 12. 3*46'46'; 16min., 7Asec. 13. 17**5r; Ihr.. 8 min., 20sec. 14. 2°,16',33.38ec.; 9min., 2.228ec. Page 302 17.1. 73*,44'59.85'' 2. 3*.2',8.25'' 3. 71°,7',45.76'' 4. 77*,3',56.7'' 18. Allegheny. 25 min., 2.948ec. Berkeley, 2hrs., 34min.. 7.638ec. Ann Arbor, 10 hrs., 26min.t 10.938ec. Chicago, Ihr., 6min., 55.79sec. Chicago, 6hrs., 59min., 47.81 sec Madison, 5hrs., 44min.. 53.73 sec. Cape G. Hope, 3hr8., 37min., 20.988ec. Paris, 4hr8., 41 min., 54.778ec. 19. 6* 15' 44.1" 38* 31' 52.96'' 156* 32' 43.96*' 16* 28' 56.86* 89* 66' 67.16* 86* 13' 25.95* 64* 20' 14.70' 70* 28' 41.65" ANSWERS 413 Page 304 1. 9 A.M; 8A.M.; 8 A.M.; 8 A.M.; 7 A.M.; 6 A.M.; 6 A.M. a.New Or., 9 hr., 10 min., 45 sec, P.M. PhUa.. 10 hr., 10 min., 45 sec, P. M. Buffalo* either of the above Washington, 10 hr., 10 min., 45 sec, P. M. Austin, 9 hr., 10 min., 45 sec, P.M. Boise City, 8 hr, 10 min., 45 sec, P.M. Los Angeles, 7 hr., 10 min., 45 sec, P. M. El Paso, 7. 8, 9 hra, 10 min.f 45 sec, P. M. Salt Lake City, 7 hrs., 10 min., 45 sec. P. M. P.M. S.NewOr. 2:16 Phila. 3:15 Buffalo 2:15 3:15 Wash. 3:15 Austin 2:16 Boise City 1:16 LosAng. 12:16 El Paso 2:15 1:15 12:15 S.L.City 12:15 Dodge ('ity, 3.- 26 A. M. Dodge City, 3:26 A. M. A.M. New Or. 4:25 Phila. 5:25 Buffalo 4:25 5:25 Wash. 5:25 Austin 4:25 Boise City 3:25 LosAng. 2:25 El Paso 3:25 2:25 1:25 S.L.City 2:25 6. 12 o'c noon; 1P.M., 4 P.M. 9 P. M.; 11 P. M. 6. 180° 7. 1 hr. Page 806 8. 12 o'c. noon; Sat. noon Page 306 9.4 A.M. Wed.; 6 A. M. Wed.; 3 A.M. Wed.; 2 A. M. Wed. lO.l. 8.25 P.M. 2. 8.25 'P.M. 3. 9.25 P.M. 4. 10.25 P.M. 5. 6.26 P. M. 6. 8.26 P.M. 7. 12.26 P. M. 8. 11.25 P.M. Page 306 2.81.60 per sq.ft.; 814.40 per sq. yd. 4. 17626 sq. ft.; 16,800; 8000 26,250; 5,200; 26,312.5 Page 307 8. 90,000 cu. ft.; 68,060; 75,000; 90,000; 126.000 10. 1. 13,333i sq. yds. 2. 13,333i sq. yds. 3. 10,000 sq. yds. 4. 11,544^ sq. yds. 5. 10,7361 sq. yds. Page 308 12. 800; 889; 1.333i; 1143 13.4* to weather, 838 . 72 5" to weather, 830.98 Cost of tin88.16 14. 31.445 16. 8113.202 16. 8220.11 17. 18,191i Page 309 19. 8445.536 20. 8202.84 21. 1728; 847.52 23. 26.880; 8168 84. 53,089i; 8742.- 26 Page 312 2. 80; 80; 80; 40; 38; 39; 40 8.8. Perry, Lot 51 No. A., 81 J. Hay, Lot 50. No. A., 82 J. Hay, E. i N. W4i. Sec. 7, No A. 80 H. Ochilltree, lot 49, No. A., 41 H. Ochill. S. E. i, N. W. i. Sec. 6, No. A. 40 H. Ochill., E. i, S. W. i, Sec 6. No. A., 80 J. Church, lot 48. No. A. 38 J. Church, lot 47. No. A. 41 Page 313 6. 32 cu. in. 6. little more than 24 cu. in. 7. 1. 76.3984 2. 269.392 3. 667.590 4. 2150.4252 5. 296.8812 ^•:7854-^- 1416 ar2 9. 761.73? gal. 10. 9.76 min. 11 14.72625 lbs. Page 314 12. 211.58 13. 562.699 14. 4 16. h 360°; 46° 18.1. 16.6 2. 188* 3. 27.2485? 4. 41.88628274 Page 316 19. 21.2142) 20. 169. 7U 21. 148.5 22. 1.80796130? 23. 27.2397321? 34. 5903.9357142? 26. 7.07J; 28.28| 28. 1257* ; 314? 29. 19704056 U sq. mi. Page 316 30. S = 47rR» 31. 120 cu. in. 82. V-2ip 33. 64 cu. in. Page 316 34. 91,636,272 cu. ft. 36. 7i?I?r< 36. 2,234,281,760 cu. ft. 37. 12.330,981,000 cu. ft. Page 317 39. 33.51 + 40. 33.51 + ; 2.807; 4. 50+; 547.- 7362911 41.Circum. mi. Moon. 6,788^ Merc, 9,522? Venus, 24,300 Earth. 24.886} Mars. 13.294? Jup.. 271,857^ Sat., 227.428^ Uran., 100,257f Nept., 109.371? Sun, 2.722,971? Surface Moon, 14.463,- 314§ Merc, 28,854,- 2671 Venus, 186,340, 000 Earth, 197,040, 66U Mars. 56,234,- 828^ Jupiter, 23,515, 642,8574 Saturn, 16,748,- 285,714? Uranus, 3,198,- 202,8571 Neptune, 3.806, 125,714? Sun, 2.359,182,- 446,714? Volume mi. Moon, 6,278,- 793.142? Merc 14,671,- 399,8541 Venus, 239.136, 333,333il Earth, 260.027, 860,52Ut Mars, 39,645,- 664,142? Jup., 339,017,- 184,523,809U Sat., 203,770,- 809,523.809^} Uranus. 17,003, 778.523,809U Nept., 22,075.- 529,142,867J Sun, 340,666,- 946,161.142.- 8571 414 RATIONAL GRAMMAR SCHOOL ARITHMETIC Pace 321 1. 5 2. 8 3. 1. 15 2. 16 3. 20 4. 36^ 6. 28 6. 30 Page 324 11. 1. 22 2. 27 3. 34 4. 25 6. 36 6. 75 7. 87 8. 97 9. 98 10. 332 14. 1. 43.6 2. 68.7 3. 71.1 4. 7.82 5. 8.71 6. 8.94 7. .74 8. .677 9. 1 . 103 Page 326 16. 1. 2.236 + 2. 2.645 + 3. 3.872 + 4. .921 + 5. .6 6. 11.180 + 7. 41.024 + 8. 82.914 + 9. 85.603 + 10. 1.359 + 11. 6.175 + 12. 8.055 + 16. 1. i ll if 4. m 5. AS 6. i*flV* 7. b* 8. 9. 85.603 + 10. 1.359 + 11. 5.175 + 12. 8.055 + 6. i"r 6. /i 7. A 8. i? 9. 19. 20. b 42.8+* 70.037+' Page 326 21. 90' 22. 4321.1 972 » 23. 38.777 24. 11248.72 sq.ft. Page 326 26. 80' 26. 133iyd8. 27. 660 ft. 28. 173.2 sq.ft. 29. 3366.36 sq. ft. 30. 804.60 sq. ft. 1. 125 cu. in.; 216 cu. in.; 343; 1331; 5832; 9- 261; 13.824 Page 327 2. 2 and 3; 3 and 4; 4 and 5; 7 and 8; 8 and 9; 9 and 10; 38 and 39; 78 and 79 3. 15.625; 300.- 763; 1953.125; 37,037,V; 6591 7.*J; I; ?; ,-o; fo; Page 330 1. 62* 2. 518 ft. 3. yes 4. 45 ft. 6. 67* ft. 6. 23f Page 331 7. 128 rods 9. r 10. 3' 11. 2200 mi. Page 332 13. 704.545Vi miles 14. 852.500 miles 16. 1700 ft. Page 333 16. 1504 ft. 17. 54 ft. 18. 846.4 Page 336 20. 4^; 4.52*; 3.78'; 5.27*; 5.35"; 8.32* 21. 280'; 450'; 512' 295': 625'; 448' 22. 3.39*; 3.6*; 3.- 425*; 3.69»; 3.- 49*; 3.125'; 3.- 395* 23. 762'; 824'; 572' 550'; 1172'; 12- 36'; U96' Page 336 26. 988 sq. ft. 26. 607.3 sq. ft..; 774; 917.2; 749 1672.5; 756.3 1. $11.90 2. 36250 3. 14.000 4. 32.800 6. $16,000 Page 337 6. 7. 8. 9. 10. 11. 12. 13. 14. 16. 16. 17. 18. 19. 20. 76%; 667.60 Page 338 1. $24087. 6Q 2. 33.874. 36i 3. $7390.512 4. $684,042.55 6. $3312133i 6. 18 mills on a dollar 7. $3001.471 8. 15 9. $158.40 10. $21,443 11. $84 Page 339 12. $100 13. .013;$13.639.11 14. $12767.48 16. 19 16. $21,354.16 1. $578.86; $672.76 Page 340 2. $143.66 3. $145.19 4. $1051.84; 6. $1040.77 6. $286,226 Page 342 1. 2.1%; $420 2. $26350; $26550 $26,200; $26400 3. $35,000 $36,060; $34.- 950; $34,960 4. Gain, $600 6. Neither loses nor gains 6. $1662.60 7. M 8. 9.7% Page 342 9. 1.85%; 1.83% 10.1.2.76%; 2.74% 2.2.76%; 2.74% 3.3.55%; 3.53% 4.2.80%; 2.85% 11. 3.69% Page 343 12. 4.66%; 4.55% 13.Wab. R. R., 6s. .42%; .41% 14. Strgt. loan. .06% 16. 14.92% ;1 122% $8860 Page 344 2. $60; $7.68 8.^ $112.61 + 9. $277.46 Page 346 10. $226.67 11. $631.24 12. $614.43 2. x + y 3. x + y 4. 80c. -45c. «= 36c. 6. m — 8 dollars 6. 45c. 7. zy dollars Page 346 8. 5A° 9. 7 rods 10. y 11. SO'-.; 40c. 12. 13. 8x; 3x' 16. 28'* 16. 12 + x; 12-y 17. 2**. 6*», 12« 18.31.50 hat; $3.00 ( :oat 19. x-$30 x-30 20. 2 Page 347 21. x-y 22. 3a 23. x-y 24. a + b 26. 3 + a miles per hr. 26. 3-a 27. 5a cents 28. ax c. 29. $40 80. 15 X 31. -S-rd.. 32. X-30 ANSWERS 415 34. c. 33. lOx +8yc m + c 3 3R. 5 + c) c. =(m + 2c) c 36. 72 xy. sq. in. (=1^") Pase 348 37. 38 3 bx + by or 6(x + y) 1. 75 lbs 3. p = 75 lbs. 3. F = 300 lbs. 4. F = 7956 lbs. 5. F» 3978 lbs. Page 349 3. 2 4. 10; 7 6. b; (b + c) lb.; (p + 36) lbs. Page 362 1. 1. 61bs. 2. 3 ft. 3. 2^ mi. 4. 4c. 5. 14 6. 21 7. 6 8. 24 9. 3 2. 6; 21; U 3. 15; 2; 19 Page 363 4. 5; 15; 25; 25 6. 4;12; 28; 16; 2 6. 9; 27; 81; 3 7. 3 8 21i 9. 4 10. 2:2 11. 7;7 12.10; 5; 3i; 2i; 1; i 14. 1. 13 2. 3 3. 80 4. 1600 5. 9 6. 89 7. 9 8. 1600 9. 39 10. 89 11. 712 12. 15 Page 363 16.1. x<y correct 2. x<y 3. x<y 4.y> —X correct 5.x + y=13 cor. 6.x-y = 6 cor. 7. (x + y)2 = 1296 8. (x-y)2 = 25 cor. 9. x2-y2<25 cor. 10.x2 + y2=97 ll.(x + y)» = x^ + 2xy + y2 12.(x-y)2 = x'-*— 2xy + y2 13.(x + y)2 = x2 + 2xy + y2cor 14.(x-y)2 = x2-2xy + y2cor 15.(x-y)2 = x2-2xy + y2 16.(x + y)(x- y) = x«-y^ 17.(x + y)(x- y) = x*-y2 18.(x-y) (X- y) = x2-y2 I9.{x-y)ix + y) = x2-y2 20.x(x + y) • = x^ + xy cor. Page 366 6. 1. 2. 3. 4. 5. 6. 7. 2 2 3 5 4 2 614 Page 367 4. 4 no. ash trees 12 no. oak trees 16 no. hickory trees 6. 15 5c. stamps 60 2 c. stamps 30 10 c. stamps 6. 20 acres 7. 6; 24; 3 Page 368 8. 9; 18; 21 9. 56,650 sq. mi. 10. 16 c. 11. 126f mi 12. 285 mi. 13. 58 mi. 14. 248 mi. 16. 369 mi. 16. Son. 315; Father, $60 17. Joseph 6; William 18 18. 12; 20 19. 30; 45 20. 40 Page 369 21. 440 rods 22. $2.50 23. 40 24. $37.50 26. 350 rods 26. 8; cir. »50| 27.R= y"^; C = 6? ^IT 1. 2. 3. 4. 6. 6. 7. 8. 9. 10. 11. 12. 13. 14. 16. 16. 17. 18. 19. 20. 3 8 7 2 24 36 20 12 3 10 3 2 2i 8 15 12 8 9 3 3 Page 360 21. Hemp cable 1" diam. W. .577 sq. in. S. .109 sq. in. S. .654 sq. in. Hemp cable 2^" diam. W. 3.60625 sq. in. L. .68125 sq. in. S. 4.0875 sq. in. 22. Tarred hemp rope, 1* diam. W. 1.036 sq. in. L. .247 sq. in. S. 1.480 sq. in. Tarred hemp 2^'* diam. W. 6.475 sq .in. L. 1.54375 sq. in. S. 9.25 sq. in. Manila rope I'' diam. W. .765 sq. in. L. .329 sq. in. S. 1.877 sq. in. Manila rope 2i* diam. W. 4.78125 sq. in. L. 2.05625 sq. in. S. 11.73125 sq. in. Page 360 23. Iron wire li* diam. W. 6.0109375 sq. in. L. 4.471875 sq. in. B. 26.58125 sq. in. Iron wire, 24* diam. W. 17.371609- 375 sq. in. L. 12.92371875 sq. in. S. 76.8198125 sq. in. 24. Steel wire, li" diam. W. 6.165625 sq. in. L. 6.9390625 sq. in. S. 43.171875 sq. in. Steel wire, 24" diam. W. 17.81865626 sq. in. L. 20.053890625 sq. in. S. 124.76671876 sq. in. Page 361 2. 12; 6 3. 4. 6. 26 girls;: 6. 125; 50 7. 24; 16 8. 15i; 7i 9. 12i; 7i 11. 7; 21 12. 371.25; 41.25 Page 362 13. 3 tin; 12 copper 14. 224tons copper 2i tons tin. 16. 2034; 28 leu. in. 16. 7; 2 17. 8; 5 18. 3; 7 19. 15 in.; 9 in. 20. 24; 8 21. 16; 12 22. 24; 40 23. I 26. 2; 12 Page 363 26. 2i; 3^ 27. -10; -35 28. 2; 3 29. x=12; y = 5 30. x = 14 y»3 31. x = 10 y = 6 if'^ 416 RATIONAL GRAMMAR SCHOOL ARITHMETIC 82. S3. 84. 85. 86. y = 5 y-14 x-1 ii b=-2H cl«2f Pace 864 1.212°F; 100**C.; SO'^R. 2.32°F.; 0**C;0**R. 8. 80*»R.; 100°C.; 180°F. Page 867 l.Ice, 32*»F.; 0*»R. Benaol, 39.9°F.; 3.52«R. Tallow, 109.4° F.; 34.4*» R. Parraffin, 114.- 8**F.; 36.8*» R. Wax. 143.6° F; 49.6* R. Sulphur. 239° F.; 92°R. Tin, 446°F.; 184°R. Bismuth. 482° F.; 200°R. Cadmium. 608° F.; 256°R. Lead, 618.8° F.; 260.8°R. Page 867 Zinc, 773.6°F.; 329. 6°R. Antimony, 829- .6°F.; 346.6°R. Silver, 1832°F.; 800°R. Copper, 2012° F.; 880°R. Gold, 2192°F.; 960°R. Caflt-lron, 2192° F.; 960°R. Cast-Steel, 2507°F.; 1100° R. Wrought-Iron, 291 2°F.; 1280° R. Platinum, 3227° F.; 1420°R. Iridium, 3642°F.; 1560° R. Page 868 2. Ether, 35°C.; 28°R. Carbon Dis., 46°C.; 36.8° R. Sulph. Acid, -10°C.; -8.° R. Chloroform, 61°C.; 48.8°R. Alcohol, 78°C.; 62.4°R. Page 868 Water, 100°C.; 80. °R. Mercury, 357° C; 285.6°R. Zinc, 1040°C.; 832°R. 4. 98.96° Fahr. 29.76° Ream. 6.Ether vapor, 384.8° Fahr. Carbonic acid, 87.8° Fahr. EtheUne, 48.2° Fahr. Oxygen, 180.4° Fahr. Nitrocen, 229° Fahr. Hydrogen, 281.2° Fahr, Page 870 8. 8114.5625 8. 8338.125 4. 5.5; 2.40625 sq. in. 6. 254.57 6. 48.43"^ Page 871 2. 8156.45 Page 272 2. 1. 884 2 396 3. 1591 4. 2436 5. 3675 Page 872 6. 8036 7. 4899 8. 8019 9. 8075 8. 1. 1296 2. 2209 3. 4096 4. 3364 5. 361 6. 9025 7. 15625 8. 21904 Page 878 6.1. 114.43 2. 20159.81 3. 8.56 4. 4780.30 7. 1. 248.729 2. 496.885 3. 1977.279 4. 4035.960 2. 1. 568.75 2. 73.27 3. 42.51 4. 2.39 Page 874 2. 1. 80.20 2. 196.54 3. 93.62 4. 1. 34 2. 212 3. 13.2 4. 222 5. 234 6. 638